#305766

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
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

603 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:

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

Polycyclic presentation:

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

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, 212962);