#7020710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fgi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6927906);