#7508512

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
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	#31413	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,26,30)(20,27,29,24)(25,28,32,31);(1,3)(2,6)(4,7)(5,8)(17,19)(18,23)(20,28)(21,26)(22,30)(24,25)(27,32)(29,31);(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,32)(28,30);(2,7)(3,8)(10,14)(13,15)(17,18,21,22)(19,23,26,30)(20,24,29,27)(25,31,32,28);(1,4,5,6)(2,3,7,8)(20,29)(24,27)(25,32)(28,31)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7415708);