#402376

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

1100 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)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,24,30)(22,26,27,33)(25,31,32,36)(28,34,29,35);(1,2,3,4)(15,19)(17,20)(22,27)(26,33)(28,29)(34,35);(5,8,9,7)(6,10,11,12)(22,28)(25,32)(26,34)(27,29)(31,36)(33,35)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=fi,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 309572);