#7602364

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(17,18,22,28)(19,30,33,24)(20,34,38,44)(21,29,23,27)(25,37,45,39)(26,40,46,32)(31,43,36,42)(35,47,48,41);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,35)(21,32)(22,33)(23,40)(25,42)(26,27)(28,30)(29,46)(31,39)(34,41)(36,37)(38,48)(43,45)(44,47);(1,3,4,7)(2,6,5,8)(17,20,21,37,22,38,23,39)(18,25,27,44,28,45,29,34)(19,31,40,48,33,36,32,35)(24,41,46,43,30,47,26,42);(11,16)(13,15)(17,19,21,40,22,33,23,32)(18,26,27,24,28,46,29,30)(20,36,37,35,38,31,39,48)(25,43,44,47,45,42,34,41);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,32,33,40)(20,37,38,39)(24,26,30,46)(25,44,45,34)(31,35,36,48)(41,42,47,43)

Polycyclic presentation:

2^9|a^2=gh,c^2=fh,d^2=gi,e^2=i,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, 7509560);