#7602838

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#25106	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,30,20,25)(21,32,31,26)(22,29,24,27);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,25,28,30)(21,22,31,24)(26,29,32,27);(10,14)(11,16)(17,20,22,31,23,19,24,21)(18,26,27,25,28,32,29,30);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,19,23,31,22,20)(18,25,29,26,28,30,27,32);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,31,20,21)(25,26,30,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=fg,e^2=i,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,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, 7510034);