#6771712

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	#25087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6678908);