#494426

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401622);