#494845

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

772 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)(5,6,10,16,11,17,27,33,12,7,13,19,9,15,25,31)(8,20,23,35,24,14,28,29,21,30,26,36,22,18,34,32);(5,7,10,16,11,15,27,33,12,6,13,19,9,17,25,31)(8,20,23,36,24,14,28,32,21,30,26,35,22,18,34,29);(2,4)(5,8)(6,14,7,18)(9,22)(10,26)(11,24)(12,21)(13,23)(15,20,17,30)(16,32,19,29)(25,28)(27,34)(31,36,33,35);(8,21)(14,18)(20,30)(22,24)(23,26)(28,34)(29,32)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402041);