#6436110

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433049
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(19,23)(20,30)(21,34)(22,25)(24,33)(26,35)(27,41)(28,37)(29,44)(31,45)(32,46)(36,42)(38,43)(39,47)(40,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,39)(25,37)(26,38)(27,32)(30,45)(33,47)(34,44)(35,43)(36,40)(41,46)(42,48);(10,13)(14,16)(17,20,21,33,22,32,35,48)(18,24,26,30,25,40,34,46)(19,27,29,42,28,31,43,47)(23,36,38,41,37,39,44,45);(9,11,12,15)(10,14,13,16)(18,25)(19,28)(20,32)(26,34)(29,43)(33,48)(36,39)(41,45);(1,3,4,7)(2,6,5,8)(19,28)(23,37)(27,31)(29,43)(36,39)(38,44)(41,45)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343306);