#287053

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,16,13,18,12,17)(11,15,19,24,20,22,21,23)(25,26,29,33,28,32,35,31)(27,34,37,40,36,30,38,39);(2,5)(6,7)(9,11,12,21,13,20,14,19)(10,15,17,23,18,22,16,24)(25,27,35,38,28,36,29,37)(26,30,31,40,32,34,33,39);(10,16)(11,19)(12,14)(15,22)(17,18)(20,21)(26,31)(27,36)(29,35)(30,39)(32,33)(34,40);(1,3,4,8)(2,6,5,7)(11,20)(15,22)(19,21)(23,24)(26,32)(27,36)(31,33)(37,38)

Polycyclic presentation:

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

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, 194249);