#237860

order	512
nilpotency class	3
isoclinism	isoclinic class of #232253
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 145056);