#6745714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2273 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,27,28,40)(20,31,32,25)(22,35,34,26)(24,36,37,30)(29,43,42,41)(33,39,47,46)(38,44,45,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(25,31)(26,35)(29,42)(30,36)(39,46)(44,48);(11,15)(13,16)(17,19,21,28)(18,24,23,37)(20,29,32,42)(22,36,34,30)(25,38,31,45)(26,40,35,27)(33,48,47,44)(39,43,46,41);(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)(20,32)(25,31)(29,42)(33,47)(38,45)(39,46)(41,43)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6652910);