#381227

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1412 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fi,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, 288423);