#448116

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fgi,c^2=fg,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355312);