#410752

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

618 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)(5,6,9,16,11,15,10,14)(7,17,19,28,20,12,25,27)(8,13,23,30,22,31,24,32)(18,33,21,35,34,26,36,29);(5,7,10,19,11,20,9,25)(6,12,14,27,15,17,16,28)(8,21,24,34,22,36,23,18)(13,29,32,33,31,35,30,26);(2,4)(5,8,9,23,11,22,10,24)(6,13,16,30,15,31,14,32)(7,18,19,21,20,34,25,36)(12,26,27,29,17,33,28,35);(6,14,15,16)(8,22)(12,27,17,28)(13,30,31,32)(18,34)(21,36)(23,24)(26,35,33,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317948);