#494425

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×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,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, 401621);