#309966

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	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,16,12,20)(13,22,23,17)(15,21,24,19)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(9,11,14,12)(10,15,18,24)(13,21,23,19)(16,17,20,22)(25,27,29,35)(26,30,32,39)(28,33,36,34)(31,37,40,38);(2,5)(6,8)(9,12,14,11)(10,16,18,20)(13,21,23,19)(15,17,24,22)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(1,3,4,7)(2,6,5,8)(10,17)(13,23)(15,20)(16,24)(18,22)(19,21)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=fh,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, 217162);