#146883

order	512
nilpotency class	4
isoclinism	isoclinic class of #11663
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54079);