#7603462

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

1571 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)(3,7)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18,23,27)(19,28,31,24)(20,32,30,41)(21,34,38,48)(22,39,36,26)(25,42,45,37)(29,46,47,43)(33,44,35,40);(2,6)(7,8)(10,13)(11,16)(17,19,23,31)(18,24,27,28)(20,22,30,36)(21,33,38,35)(25,43,45,46)(26,41,39,32)(29,42,47,37)(34,40,48,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,31,23,30,36,19)(18,24,26,32,27,28,39,41)(21,35,37,29,38,33,42,47)(25,44,34,46,45,40,48,43);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,36,42,23,38,22,37)(18,25,39,48,27,45,26,34)(19,29,30,35,31,47,20,33)(24,40,41,46,28,44,32,43);(2,6)(7,8)(10,13)(14,15)(17,22,23,36)(18,26,27,39)(19,30,31,20)(21,37,38,42)(24,41,28,32)(25,34,45,48)(29,35,47,33)(40,46,44,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fhi,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, 7510658);