#7596417

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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1536 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)(11,15)(12,13)(14,16)(17,18)(19,26)(20,31)(21,32)(22,27)(23,29)(24,28)(25,30);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,24)(21,23)(22,30)(26,27)(28,32)(29,31);(10,13)(14,15)(17,20,24,19,22,21,23,30)(18,25,29,32,27,26,28,31);(9,11,12,16)(10,14,13,15)(17,21,23,19,22,20,24,30)(18,26,28,32,27,25,29,31);(1,3,4,8)(2,6,5,7)(18,27)(25,26)(28,29)(31,32)

Polycyclic presentation:

2^9|c^2=h,d^2=g,e^2=fi,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

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, 7503613);