#238238

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145434);