#7596556

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
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	#59652	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1627 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7503752);