#127724

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=gi,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34920);