#6485898

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,24,28)(19,31,34,26)(20,35,33,25)(21,27,39,48)(22,30,23,29)(32,45,37,43)(36,44,38,42)(40,47,41,46);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,23,20,24,34,22,33)(18,25,30,26,28,35,29,31)(21,37,41,38,39,32,40,36)(27,44,47,45,48,42,46,43);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,23,34,24,33,22,19)(18,26,30,35,28,31,29,25)(21,38,41,32,39,36,40,37)(27,45,47,42,48,43,46,44);(9,11,13,12)(10,15,14,16)(17,21)(18,27)(19,32)(20,36)(22,40)(23,41)(24,39)(25,42)(26,43)(28,48)(29,46)(30,47)(31,45)(33,38)(34,37)(35,44);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(18,28)(21,39)(25,35)(26,31)(29,30)(32,37)(36,38)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6393094);