#335966

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=eh,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 243162);