#6635556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6542752);