#6556212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 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)(4,5)(7,8)(17,18)(19,28)(20,34)(21,25)(22,27)(23,31)(24,36)(26,29)(30,44)(32,40)(33,39)(35,42)(37,41)(38,47)(43,46)(45,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,29,32,22,33,21,31)(18,23,25,39,27,40,26,28)(20,30,35,45,37,47,36,46)(24,38,41,48,42,44,34,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,36,22,37,29,35)(18,24,26,34,27,42,25,41)(19,30,31,46,33,47,32,45)(23,38,28,43,40,44,39,48);(9,11)(10,14)(12,15)(13,16)(17,21,22,29)(18,25,27,26)(19,31,33,32)(20,35,37,36)(23,39,40,28)(24,34,42,41)(30,45,47,46)(38,43,44,48);(11,15)(14,16)(17,21,22,29)(18,25,27,26)(19,32,33,31)(20,36,37,35)(23,28,40,39)(24,41,42,34)(30,45,47,46)(38,43,44,48)

Polycyclic presentation:

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

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, 6463408);