#6771002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1641 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6678198);