#462118

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
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

583 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)(5,6,10,17,12,18,11,16)(7,15,9,13,21,29,20,30)(8,22,26,33,25,32,24,14)(19,28,27,36,23,31,35,34);(5,7,11,20,12,21,10,9)(6,13,17,29,18,30,16,15)(8,23,24,27,25,19,26,35)(14,28,22,36,33,31,32,34);(5,8,11,24,12,25,10,26)(6,14,17,22,18,33,16,32)(7,19,20,35,21,23,9,27)(13,28,29,36,30,31,15,34);(2,4)(5,9,10,21,12,20,11,7)(6,15,16,30,18,29,17,13)(8,23,24,27,25,19,26,35)(14,31,22,34,33,28,32,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^c=hi

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, 369314);