#6469369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,22,24)(19,27,29,43)(20,25,33,39)(21,35,28,26)(23,36,38,31)(30,42,32,46)(34,48,44,41)(37,47,40,45);(9,10)(11,16)(12,14)(13,15)(17,19,28,36,22,29,21,31)(18,23,35,43,24,38,26,27)(20,32,44,45,33,30,34,47)(25,40,48,42,39,37,41,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,24)(19,29)(20,33)(26,35)(31,36)(34,44)(37,40)(42,46);(10,14)(13,16)(17,20)(18,25)(19,30)(21,34)(22,33)(23,37)(24,39)(26,41)(27,42)(28,44)(29,32)(31,45)(35,48)(36,47)(38,40)(43,46);(1,4,5,3)(2,6,7,8)(20,33)(25,39)(30,32)(34,44)(37,40)(41,48)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376565);