#7508249

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(18,23)(19,27)(25,26)(29,30);(1,3)(2,6)(4,7)(5,8)(17,18,21,23)(19,28,27,32)(20,26,22,25)(24,30,31,29);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,27)(22,30)(23,31)(25,28)(26,32);(2,7)(3,8)(10,14)(13,15)(17,20,21,22)(18,25,23,26)(19,29,27,30)(24,28,31,32);(1,4,5,6)(2,3,7,8)(19,27)(24,31)(28,32)(29,30)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7415445);