#7602910

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=fg,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^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, 7510106);