#6745886

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2226 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)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,34)(28,40)(29,43)(33,39)(36,37)(38,48)(41,42)(44,45)(46,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(20,32)(26,35)(30,36)(33,47)(38,45)(41,43);(11,15)(13,16)(17,19)(18,24)(20,29)(21,28)(22,36)(23,37)(25,38)(26,40)(27,35)(30,34)(31,45)(32,42)(33,48)(39,43)(41,46)(44,47);(2,6)(5,7)(17,20,21,32)(18,25,23,31)(19,29,28,42)(22,33,34,47)(24,38,37,45)(26,39,35,46)(27,41,40,43)(30,44,36,48);(3,8)(5,7)(11,15)(13,16)(19,28)(20,32)(24,37)(25,31)(27,40)(30,36)(33,47)(39,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6653082);