#264170

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171366);