#348851

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=ei,c^2=hi,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=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, 256047);