#6485456

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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

1483 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)(17,18,23,27)(19,28,32,45)(20,25,22,37)(21,26,36,43)(24,29,39,31)(30,44,34,48)(33,40,35,42)(38,46,41,47);(9,10)(11,16)(12,15)(13,14)(17,19,22,31,23,32,20,29)(18,24,25,28,27,39,37,45)(21,34,35,46,36,30,33,47)(26,41,40,48,43,38,42,44);(2,5)(6,8)(10,14)(15,16)(17,20,23,22)(18,25,27,37)(19,29,32,31)(21,35,36,33)(24,28,39,45)(26,42,43,40)(30,47,34,46)(38,48,41,44);(9,11,13,12)(10,15,14,16)(17,21,23,36)(18,26,27,43)(19,30,32,34)(20,33,22,35)(24,38,39,41)(25,40,37,42)(28,44,45,48)(29,46,31,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(21,36)(26,43)(30,34)(33,35)(38,41)(40,42)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,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, 6392652);