#6651490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,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, 6558686);