#6966453

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(3,7)(4,6)(5,8)(17,18,23,27,22,28,21,35)(19,29,31,40,32,24,36,41)(20,26,33,43,34,42,37,25)(30,45,48,38,47,39,46,44);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,39)(27,41)(28,29)(33,48)(34,47)(35,40)(37,46)(42,45)(43,44);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,34)(18,25,28,43)(19,30,32,47)(21,33,23,37)(24,38,29,44)(26,27,42,35)(31,46,36,48)(39,41,45,40);(2,6)(5,7)(10,13)(14,16)(17,20)(18,26)(19,30)(21,37)(22,34)(23,33)(24,39)(25,35)(27,43)(28,42)(29,45)(31,48)(32,47)(36,46)(38,40)(41,44);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,28,35)(19,31,32,36)(20,33,34,37)(24,40,29,41)(25,42,43,26)(30,46,47,48)(38,39,44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873649);