#6801461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708657);