#7620635

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgh,e^2=g,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

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, 7527831);