#511904

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2193	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

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, 419100);