#6469187

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1536 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)(17,18,23,36,22,25,21,27)(19,28,35,24,29,43,31,39)(20,26,37,48,33,40,34,42)(30,44,47,38,32,46,45,41);(9,10)(11,16)(12,14)(13,15)(17,19)(18,24)(20,32)(21,35)(22,29)(23,31)(25,39)(26,41)(27,43)(28,36)(30,33)(34,45)(37,47)(38,40)(42,44)(46,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,25)(19,29)(20,33)(27,36)(31,35)(34,37)(38,41)(44,46);(10,14)(13,16)(17,20)(18,26)(19,30)(21,34)(22,33)(23,37)(24,38)(25,40)(27,42)(28,44)(29,32)(31,45)(35,47)(36,48)(39,41)(43,46);(1,4,5,3)(2,6,7,8)(20,33)(26,40)(30,32)(34,37)(38,41)(42,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6376383);