#237886

order	512
nilpotency class	3
isoclinism	isoclinic class of #232331
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=hi,d^2=hi,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=dhi,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145082);