#7597182

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,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,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, 7504378);