#6856512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6763708);