#512923

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260454, #72322412, #72322434
maximal normal subgroup of	#72260454, #72322412, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420119);