#7596616

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45388	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,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,d^c=di,e^c=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, 7503812);