#6435859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=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, 6343055);