#7604147

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1276 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)(9,10,13,12)(11,14,16,15)(17,18)(19,22)(20,27)(21,25)(23,26)(24,30)(28,31)(29,32);(2,5)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,23,25)(18,22,26,21)(20,28,29,24)(27,31,32,30);(9,12,13,10)(11,15,16,14)(17,20,22,31,23,29,21,30)(18,24,25,32,26,28,19,27);(11,16)(14,15)(17,19,22,18,23,25,21,26)(20,27,31,24,29,32,30,28);(1,3,4,8)(2,6,5,7)(20,29)(24,28)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,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,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, 7511343);