#494956

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 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)(5,6,11,7,10,15,26,19,12,16,28,20,27,14,9,18)(8,13,23,17,22,31,21,34,24,32,35,29,36,30,25,33);(5,7,11,15,10,19,26,16,12,20,28,14,27,18,9,6)(8,17,23,31,22,34,21,32,24,29,35,30,36,33,25,13);(2,4)(5,8)(6,13)(7,17)(9,25)(10,22)(11,23)(12,24)(14,30)(15,31)(16,32)(18,33)(19,34)(20,29)(21,26)(27,36)(28,35);(5,9,10,11,12,26,27,28)(6,7,14,18,16,20,15,19)(8,21,22,35,24,25,36,23)(13,29,30,34,32,17,31,33)

Polycyclic presentation:

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

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, 402152);