#417213

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fg,d^2=h,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, 324409);