#6434878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18,23,29)(19,24,32,40)(20,26,34,42)(21,28,22,27)(25,36,41,33)(30,38,46,47)(31,43,35,39)(37,44,48,45);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,32)(18,24,29,40)(20,30,34,46)(21,35,22,31)(25,37,41,48)(26,38,42,47)(27,43,28,39)(33,45,36,44);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,33)(22,36)(23,34)(24,37)(26,27)(28,42)(29,41)(31,44)(32,46)(35,45)(38,43)(39,47)(40,48);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,15,14,16)(17,20)(18,26)(19,30)(21,36)(22,33)(23,34)(24,38)(25,27)(28,41)(29,42)(31,45)(32,46)(35,44)(37,43)(39,48)(40,47);(17,21,23,22)(18,27,29,28)(19,31,32,35)(20,33,34,36)(24,39,40,43)(25,26,41,42)(30,44,46,45)(37,47,48,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,e^b=ei,g^c=gi,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, 6342074);