#6865814

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	#24162	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,33)(21,28)(22,24)(23,27)(25,32)(26,35)(30,42)(31,45)(34,40)(36,41)(37,38)(39,46)(43,44)(47,48);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(18,24)(19,30)(26,33)(27,28)(31,44)(32,37)(40,41)(46,47);(11,16)(13,15)(17,19,22,30)(18,25,24,38)(20,31,35,44)(21,32,23,37)(26,39,33,48)(27,29,28,42)(34,46,36,47)(40,45,41,43);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20)(18,26)(19,31)(21,36)(22,35)(23,34)(24,33)(25,39)(27,41)(28,40)(29,43)(30,44)(32,47)(37,46)(38,48)(42,45);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,27,24,28)(19,32,30,37)(20,34,35,36)(25,29,38,42)(26,40,33,41)(31,46,44,47)(39,45,48,43)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6773010);