#6797941

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,24)(19,28,29,41)(20,32,35,26)(21,27,23,38)(25,36,39,31)(30,46,33,45)(34,44,37,43)(40,48,42,47);(1,3,4,8)(2,5,6,7)(11,12)(14,16)(18,24)(19,29)(26,32)(27,38)(30,33)(31,36)(43,44)(47,48);(3,8)(5,7)(9,11,13,12)(10,14,15,16)(17,19,21,36,22,29,23,31)(18,25,38,41,24,39,27,28)(20,33,37,48,35,30,34,47)(26,42,43,45,32,40,44,46);(2,6)(5,7)(17,20)(18,26)(19,30)(21,37)(22,35)(23,34)(24,32)(25,40)(27,44)(28,45)(29,33)(31,48)(36,47)(38,43)(39,42)(41,46);(9,12,13,11)(10,14,15,16)(17,21,22,23)(18,27,24,38)(19,31,29,36)(20,34,35,37)(25,41,39,28)(26,43,32,44)(30,47,33,48)(40,45,42,46)

Polycyclic presentation:

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

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, 6705137);