#435454

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=ef,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=dei,e^d=ei,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, 342650);