#6915422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822618);