#6386344

order	512
nilpotency class	3
isoclinism	isoclinic class of #61591
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3480	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3137 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,22,23,21,26,19,25)(20,28,30,24,29,31,27,32);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,31,32,28);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,21,29)(18,24,26,32)(19,27,22,30)(23,31,25,28);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,19,21,22)(18,25,26,23)(20,27,29,30)(24,28,32,31);(2,10)(7,11)(8,9)(15,16)(18,26)(23,25)(24,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293540);