#6456200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,30)(24,27)(28,32)(29,31);(2,6)(7,8)(17,19,23,18)(20,26,32,29)(21,25,31,28)(22,30,24,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,32)(26,27)(29,30);(1,4,5,3)(2,8,6,7)(17,21,23,31)(18,26,19,29)(20,24,32,22)(25,27,28,30);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,19,30)(20,31,32,21)(25,26,28,29)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6363396);