#6798274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,34)(28,40)(29,43)(33,38)(36,42)(37,48)(39,41)(44,45)(46,47);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(18,23)(19,28)(25,31)(26,35)(29,41)(30,42)(38,46)(44,48);(10,13)(14,16)(17,19,22,30,21,28,34,42)(18,24,26,27,23,36,35,40)(20,29,33,44,32,41,47,48)(25,37,38,39,31,45,46,43);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,20,21,32)(18,25,23,31)(19,29,28,41)(22,33,34,47)(24,37,36,45)(26,38,35,46)(27,39,40,43)(30,44,42,48);(1,5,6,4)(2,7,3,8)(20,32)(25,31)(29,41)(33,47)(37,45)(38,46)(39,43)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6705470);