#297332

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204528);