#6468156

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1526 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375352);