#7603923

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#25380	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,16,12,15)(17,18,23,28)(19,26,20,24)(21,25,30,32)(22,31,29,27);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,20)(18,24,28,26)(21,29,30,22)(25,31,32,27);(10,14)(15,16)(17,20,22,21,23,19,29,30)(18,25,27,24,28,32,31,26);(9,11,13,12)(10,15,14,16)(17,21,22,19,23,30,29,20)(18,26,27,25,28,24,31,32);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(17,22,23,29)(18,27,28,31)(19,21,20,30)(24,25,26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=g,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,d^c=di,e^c=ei,e^d=ei

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, 7511119);