#7620668

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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1253 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgh,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7527864);