#7602375

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,28,30,24)(20,31,33,45)(21,34,37,48)(23,39,36,27)(25,41,44,38)(29,46,47,42)(32,43,35,40);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,26,28)(20,23,33,36)(21,32,37,35)(25,42,44,46)(27,45,39,31)(29,41,47,38)(34,40,48,43);(1,3,4,7)(2,6,5,8)(17,20,23,30,22,33,36,19)(18,24,27,31,26,28,39,45)(21,35,38,29,37,32,41,47)(25,43,34,46,44,40,48,42);(11,16)(13,15)(17,21,36,41,22,37,23,38)(18,25,39,48,26,44,27,34)(19,29,33,35,30,47,20,32)(24,40,45,46,28,43,31,42);(2,5)(6,8)(10,14)(13,15)(18,26)(21,37)(24,28)(27,39)(29,47)(31,45)(32,35)(38,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fhi,d^2=gi,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^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, 7509571);