#6485331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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

1567 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)(9,10)(11,15)(12,16)(13,14)(17,18,21,26,22,27,35,43)(19,28,30,38,31,42,34,23)(20,24,33,39,32,25,40,41)(29,44,47,37,46,45,48,36);(9,10)(11,16)(12,15)(13,14)(17,19,22,31)(18,23,27,38)(20,29,32,46)(21,34,35,30)(24,36,25,37)(26,42,43,28)(33,48,40,47)(39,45,41,44);(2,5)(6,8)(10,14)(15,16)(17,20,22,32)(18,24,27,25)(19,29,31,46)(21,33,35,40)(23,36,38,37)(26,39,43,41)(28,44,42,45)(30,47,34,48);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,27)(26,41)(28,45)(30,47)(31,46)(34,48)(35,40)(36,38)(39,43)(42,44);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(20,32)(24,25)(29,46)(33,40)(36,37)(39,41)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6392527);