#391817

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,9)(8,10)(11,12)(14,15)(18,19)(21,22)(23,32)(24,28)(25,29)(26,30)(27,31)(33,36)(34,35);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,23,27,34,25,24,26,33)(22,28,31,36,29,32,30,35);(2,4)(6,10)(9,12)(13,15,17,14)(16,18,20,19)(21,24,25,23)(22,28,29,32)(26,33,27,34)(30,36,31,35);(7,11)(9,12)(13,16,17,20)(14,18,15,19)(23,24)(28,32)(33,34)(35,36)

Polycyclic presentation:

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

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, 299013);