#6495521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470355
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

1455 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)(11,15)(14,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,29)(26,35)(28,41)(30,42)(31,46)(32,39)(33,48)(36,44)(37,45)(38,47)(40,43);(9,10)(11,14)(12,13)(15,16)(17,19,21,29,22,30,35,45)(18,23,26,27,25,37,34,42)(20,31,33,47,32,28,43,44)(24,38,40,46,39,36,48,41);(2,5)(6,8)(10,13)(14,16)(17,20,22,32)(18,24,25,39)(19,28,30,31)(21,33,35,43)(23,36,37,38)(26,40,34,48)(27,41,42,46)(29,44,45,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,32)(24,39)(28,31)(33,43)(36,38)(40,48)(41,46)(44,47);(9,11,12,15)(10,14,13,16)(18,25)(23,37)(24,39)(26,34)(27,42)(36,38)(40,48)(41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402717);