#6914647

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30503	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(17,18)(19,28)(20,33)(21,27)(22,24)(23,37)(25,32)(26,34)(29,43)(30,45)(31,38)(35,41)(36,40)(39,47)(42,44)(46,48);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,34)(27,37)(31,32)(35,36)(39,48)(42,45);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,31,22,29,23,32)(18,25,37,43,24,38,27,28)(20,30,36,47,34,44,35,46)(26,39,40,45,33,48,41,42);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,33)(25,39)(27,41)(28,42)(29,44)(31,47)(32,46)(37,40)(38,48)(43,45);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,24,37)(19,31,29,32)(20,35,34,36)(25,28,38,43)(26,40,33,41)(30,46,44,47)(39,45,48,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6821843);