#7089626

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

1327 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=h,d^2=fi,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, 6996822);