#6557245

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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,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, 6464441);