#348868

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
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

856 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)(5,6,8,11)(7,15,16,20)(9,12,10,14)(13,17,19,18)(24,31)(25,33)(27,35)(28,30)(29,36)(32,34);(1,2,3,4)(6,11)(7,16)(12,14)(17,18)(21,22,26,23)(24,27,32,29)(25,28,33,30)(31,35,34,36);(6,12)(9,10)(11,14)(13,20)(15,19)(17,18)(21,23,26,22)(24,29,32,27)(25,28,33,30)(31,35,34,36);(5,7,8,16)(6,13,11,19)(9,18,10,17)(12,20,14,15)(21,24,26,32)(22,27,23,29)(25,34,33,31)(28,36,30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256064);