#326059

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

587 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,15)(13,22)(14,18)(16,20)(17,23)(21,24)(25,26)(27,33)(28,31)(29,32)(30,36)(34,38)(35,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11,13,12,14,20,23,21)(10,15,17,19,18,24,22,16)(25,27,34,35,28,36,29,37)(26,30,38,40,31,33,32,39);(1,4,5,3)(2,6,7,8)(9,11,13,12,14,20,23,21)(10,15,17,19,18,24,22,16)(26,31)(27,35)(29,34)(30,39)(33,40)(36,37);(2,7)(6,8)(9,12,13,20,14,21,23,11)(10,16,17,15,18,19,22,24)(26,31)(27,36)(32,38)(35,37)

Polycyclic presentation:

2^9|b^2=eg,c^2=ei,d^2=fi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dhi,d^c=dh,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, 233255);