#6386120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342764
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	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293316);