#391909

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi

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, 299105);