#7080742

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,29,32,41,30,25,33,39)(20,34,37,44,38,26,35,40)(31,42,45,47,36,43,46,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,35)(21,23)(25,41)(27,28)(29,39)(31,45)(36,46)(37,38)(40,44)(42,43);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,36)(21,33)(22,30)(23,32)(24,39)(26,43)(27,41)(28,29)(31,38)(34,42)(35,45)(37,46)(40,48)(44,47);(3,8)(5,7)(11,16)(13,15)(17,20,21,35,22,38,23,37)(18,26,27,44,28,34,24,40)(19,31,33,46,30,36,32,45)(25,42,41,48,29,43,39,47);(10,14)(13,15)(17,21,22,23)(18,24,28,27)(19,32,30,33)(20,37,38,35)(25,41,29,39)(26,44,34,40)(31,46,36,45)(42,47,43,48)

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6987938);