#494074

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

678 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:

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

Polycyclic presentation:

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

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, 401270);