#348943

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

911 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256139);