#347534

order	512
nilpotency class	3
isoclinism	isoclinic class of #346294
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1000 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=h,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254730);