#7007755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914951);