#6386111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3490	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293307);