#146868

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
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×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	C₄×#99
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

680 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)(7,11)(8,13,19,12)(9,14)(10,15)(16,23,29,22)(17,24)(18,25)(20,27,32,26)(21,28)(30,34,36,33)(31,35);(1,2,4,3)(5,7,9,17,10,18,21,31)(6,11,14,24,15,25,28,35)(8,16,20,30,19,29,32,36)(12,22,26,33,13,23,27,34);(1,3,4,2)(6,12)(8,19)(11,22)(13,15)(14,26)(16,29)(20,32)(23,25)(24,33)(27,28)(30,36)(34,35)

Polycyclic presentation:

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

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, 54064);