#391778

order	512
nilpotency class	3
isoclinism	isoclinic class of #213537
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1672 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,27,28,25)(21,26)(23,31,32,29)(24,30);(3,7)(6,8)(10,14)(13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31);(1,3,4,7)(2,6,5,8)(11,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298974);