#6434232

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(17,18,22,27,24,29,23,28)(19,30,32,40,20,34,33,25)(21,26,37,42,36,41,38,43)(31,44,45,48,35,47,46,39);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,20)(18,25,29,40)(21,31,36,35)(22,33,23,32)(26,39,41,48)(27,34,28,30)(37,46,38,45)(42,47,43,44);(11,15)(13,16)(17,20,24,19)(18,25,29,40)(21,35,36,31)(22,32,23,33)(26,39,41,48)(27,34,28,30)(37,45,38,46)(42,47,43,44);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21)(18,26)(19,31)(20,35)(22,37)(23,38)(24,36)(25,39)(27,42)(28,43)(29,41)(30,44)(32,45)(33,46)(34,47)(40,48);(21,36)(26,41)(31,35)(37,38)(39,48)(42,43)(44,47)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,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, 6341428);