#7504685

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2231 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:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7411881);