#6865829

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	#31001	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)(17,18,23,29)(19,31,35,25)(20,27,21,37)(22,30,24,28)(26,38,43,36)(32,46,33,41)(34,42,39,44)(40,48,45,47);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,32)(21,33)(22,39)(23,35)(24,34)(26,40)(27,41)(28,44)(29,31)(30,42)(36,48)(37,46)(38,47)(43,45);(11,16)(13,15)(17,20,23,21)(18,26,29,43)(19,32,35,33)(22,36,24,38)(25,40,31,45)(27,30,37,28)(34,47,39,48)(41,42,46,44);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,23,20)(18,27,29,37)(19,33,35,32)(22,36,24,38)(25,41,31,46)(26,30,43,28)(34,47,39,48)(40,42,45,44);(1,5,6,4)(2,7,3,8)(17,22,23,24)(18,28,29,30)(19,34,35,39)(20,36,21,38)(25,42,31,44)(26,27,43,37)(32,47,33,48)(40,46,45,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fh,e^2=fh,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, 6773025);