#6468644

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(10,14)(15,16)(17,18,23,37,22,27,21,26)(19,28,36,24,30,43,31,40)(20,32,38,42,34,25,35,47)(29,45,48,41,33,44,46,39);(9,10,13,14)(11,16,12,15)(17,19,22,30)(18,24,27,40)(20,33,34,29)(21,36,23,31)(25,41,32,39)(26,43,37,28)(35,46,38,48)(42,44,47,45);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,22,34)(18,25,27,32)(19,29,30,33)(21,35,23,38)(24,39,40,41)(26,42,37,47)(28,44,43,45)(31,46,36,48);(9,12,13,11)(10,16,14,15)(19,30)(24,40)(28,43)(29,33)(31,36)(39,41)(44,45)(46,48);(1,3,4,7)(2,6,5,8)(20,34)(25,32)(29,33)(35,38)(39,41)(42,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375840);