#511759

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3074 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418955);