#6501921

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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)(4,8)(5,6)(17,18,23,28)(19,24,32,41)(20,33,21,26)(22,36,37,27)(25,35,43,34)(29,44,30,39)(31,47,48,40)(38,46,42,45);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,41)(20,30,21,29)(22,31,37,48)(25,42,43,38)(26,44,33,39)(27,40,36,47)(34,46,35,45);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,23)(22,35)(24,38)(26,27)(28,43)(30,32)(31,46)(33,36)(34,37)(39,40)(41,42)(44,47)(45,48);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,21,23,20)(18,26,28,33)(19,30,32,29)(22,35,37,34)(24,39,41,44)(25,36,43,27)(31,46,48,45)(38,47,42,40);(10,14)(13,15)(20,21)(25,43)(26,33)(29,30)(34,35)(38,42)(39,44)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6409117);