#6434370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,30,34,42)(20,35,21,26)(22,27,39,28)(24,38,43,33)(25,36,44,37)(31,40,32,46)(41,48,45,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,34)(18,24,29,43)(20,31,21,32)(22,38,39,33)(25,40,44,46)(26,41,35,45)(27,30,28,42)(36,48,37,47);(11,15)(13,16)(17,20,23,21)(18,25,29,44)(19,31,34,32)(22,36,39,37)(24,40,43,46)(26,27,35,28)(30,45,42,41)(33,47,38,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,20)(18,26,29,35)(19,32,34,31)(22,36,39,37)(24,41,43,45)(25,27,44,28)(30,46,42,40)(33,47,38,48);(17,22,23,39)(18,27,29,28)(19,33,34,38)(20,36,21,37)(24,42,43,30)(25,35,44,26)(31,47,32,48)(40,41,46,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 6341566);