#6452318

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,d^2=g,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, 6359514);