#435743

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
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

615 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)(5,6,10,17,12,16,11,15)(7,9,21,28,20,29,19,13)(8,22,26,33,27,14,25,32)(18,24,34,30,35,31,23,36);(5,7,10,21,12,20,11,19)(6,13,17,9,16,28,15,29)(8,23,25,35,27,34,26,18)(14,31,33,30,22,24,32,36);(5,8,10,25,12,27,11,26)(6,14,17,33,16,22,15,32)(7,18,21,23,20,35,19,34)(9,24,28,36,29,31,13,30);(2,4)(5,9,10,28,12,29,11,13)(6,7,15,19,16,20,17,21)(8,24,25,36,27,31,26,30)(14,18,32,34,22,35,33,23)

Polycyclic presentation:

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

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, 342939);