#6495030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470368
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,23,26)(19,21,25,22)(20,28,29,24)(27,31,32,30);(9,10)(11,14)(12,13)(15,16)(17,19,22,18,23,25,21,26)(20,27,31,24,29,32,30,28);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,20,23,29)(18,24,26,28)(19,27,25,32)(21,30,22,31);(10,13)(14,16)(20,29)(24,28)(27,32)(30,31);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,26,19)(20,30,29,31)(24,32,28,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^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^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, 6402226);