#6495148

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470362
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=gi,d^2=i,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

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, 6402344);