#6999067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(19,23)(20,25)(21,34)(22,26)(24,33)(27,36)(28,44)(29,38)(30,40)(31,48)(32,43)(35,42)(37,45)(39,47)(41,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,40)(20,29,32,28)(21,31,36,47)(24,41,42,37)(25,38,43,44)(27,39,34,48)(33,46,35,45);(9,11)(10,13)(12,16)(14,15)(17,20,22,32)(18,24,26,42)(19,28,30,29)(21,35,36,33)(23,37,40,41)(25,27,43,34)(31,46,47,45)(38,48,44,39);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,34)(26,43)(27,42)(28,30)(31,46)(35,36)(37,39)(40,44)(41,48)(45,47);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(19,30)(20,32)(27,34)(31,47)(33,35)(37,41)(38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=f,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=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6906263);