#417212

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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

998 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)(5,6,11,15,10,14,24,16)(7,17,20,32,19,12,21,30)(8,22,9,26,25,13,27,31)(18,28,23,29,35,33,36,34);(5,7,11,21,10,19,24,20)(6,12,15,32,14,17,16,30)(8,23,9,18,25,36,27,35)(13,29,31,28,22,34,26,33);(1,2,3,4)(5,8,24,27,10,25,11,9)(6,13,16,26,14,22,15,31)(7,18,21,36,19,35,20,23)(12,28,32,34,17,33,30,29);(5,9,11,25,10,27,24,8)(6,13,16,26,14,22,15,31)(7,18,21,36,19,35,20,23)(12,29,30,33,17,34,32,28)

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324408);