#512836

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

768 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6)(7,14)(8,15)(9,26)(10,32)(11,35)(12,18)(13,20)(16,29)(17,28)(19,40)(21,50)(22,44)(23,43)(24,54)(25,47)(27,41)(30,62)(31,58)(33,42)(34,64)(36,48)(37,55)(38,52)(39,49)(45,56)(46,61)(51,67)(53,68)(57,60)(59,65)(63,66);(5,7)(6,13)(8,19)(9,27)(10,33)(11,23)(12,22)(14,26)(15,36)(16,42)(17,41)(18,39)(20,29)(21,28)(24,55)(25,51)(30,63)(31,59)(32,37)(34,53)(35,38)(40,54)(43,62)(44,58)(45,67)(46,68)(47,65)(48,56)(49,61)(50,57)(52,60)(64,66);(2,4)(5,7)(6,14)(8,23)(9,20)(10,21)(11,19)(12,22)(13,26)(15,43)(16,41)(17,42)(18,44)(24,52)(25,53)(27,29)(28,33)(30,48)(31,49)(32,50)(34,51)(35,40)(36,62)(37,57)(38,54)(39,58)(45,66)(46,65)(47,68)(55,60)(56,63)(59,61)(64,67);(5,8,12,11)(6,15,18,35)(7,19,22,23)(9,24,31,30)(10,25,34,28)(13,36,39,38)(14,40,44,43)(16,45,46,57)(17,32,47,64)(20,48,49,52)(21,33,51,53)(26,54,58,62)(27,55,59,63)(29,56,61,60)(37,65,66,41)(42,67,68,50)

Polycyclic presentation:

2^9|d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260454, #72322420, #72322434
maximal normal subgroup of	#72260454, #72322420, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420032);