#7088092

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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,19,23,21,22,26)(20,25,29,27,31,28,30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,26)(18,22,21,24)(20,27,31,32)(25,30,28,29);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,20,23,31)(18,25,21,28)(19,27,26,32)(22,30,24,29);(11,16)(13,15)(17,21,22,19,23,18,24,26)(20,28,30,27,31,25,29,32);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,22,23,24)(18,19,21,26)(20,29,31,30)(25,32,28,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=hi,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 6995288);