#512919

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

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

normal subgroups only

This group has 485 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)(7,12)(8,13)(9,26)(10,29)(11,17)(14,22)(15,23)(16,27)(18,31)(19,30)(20,48)(21,49)(24,41)(25,37)(28,55)(32,45)(33,46)(34,44)(35,51)(36,50)(38,54)(39,52)(40,53)(42,56)(43,66)(47,62)(57,67)(58,65)(59,63)(60,64)(61,68);(1,2,3,4)(5,7)(6,12)(8,19)(9,20)(10,21)(11,18)(13,30)(14,32)(15,33)(16,34)(17,31)(22,45)(23,46)(24,47)(25,42)(26,48)(27,44)(28,43)(29,49)(35,57)(36,58)(37,56)(38,61)(39,59)(40,60)(41,62)(50,65)(51,67)(52,63)(53,64)(54,68)(55,66);(6,13)(8,22)(9,23)(10,27)(12,30)(14,15)(16,41)(17,37)(19,45)(20,46)(21,44)(24,54)(25,52)(26,35)(28,53)(29,36)(31,56)(32,33)(34,62)(38,55)(39,40)(42,63)(43,64)(47,68)(48,57)(49,58)(50,51)(59,60)(61,66)(65,67);(7,18)(12,31)(19,42)(20,43)(21,44)(30,56)(32,59)(33,60)(34,49)(45,63)(46,64)(47,65)(48,66)(57,61)(58,62)(67,68)

Polycyclic presentation:

2^9|b^2=e,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, #72322401, #72322427
maximal normal subgroup of	#72260454, #72322401, #72322427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420115);