#7079716

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	#23965	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,22,25,23,26,21,20)(19,24,29,28,27,32,31,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,20)(19,27)(21,22)(24,28)(25,26)(30,32);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,31)(22,29)(23,27)(25,28)(26,32);(3,8)(5,7)(11,16)(13,15)(17,20,22,18,23,25,21,26)(19,28,29,32,27,30,31,24);(10,14)(13,15)(17,21,23,22)(18,25,26,20)(19,29,27,31)(24,30,32,28)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=fg,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, 6986912);