#6495367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25608	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 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)(11,15)(14,16)(17,18)(19,28)(20,33)(21,25)(22,36)(23,26)(24,31)(27,37)(29,44)(30,42)(32,43)(34,41)(35,48)(38,46)(39,47)(40,45);(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,32,37,47)(18,24,27,28,26,39,36,43)(20,30,34,46,21,29,35,45)(25,40,41,44,33,38,48,42);(2,5)(6,8)(10,13)(14,16)(17,20,23,21)(18,25,26,33)(19,29,32,30)(22,34,37,35)(24,38,39,40)(27,41,36,48)(28,42,43,44)(31,45,47,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,23,20)(18,25,26,33)(19,30,32,29)(22,35,37,34)(24,38,39,40)(27,41,36,48)(28,42,43,44)(31,46,47,45);(9,11,12,15)(10,14,13,16)(18,26)(20,21)(24,39)(27,36)(28,43)(29,30)(34,35)(45,46)

Polycyclic presentation:

2^9|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=dh,e^c=ei,e^d=ei

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, 6402563);