#258358

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,13,21)(12,19,20,23)(16,22,24,18)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(2,5)(6,7)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19)(25,27,30,37)(26,31,34,40)(28,36,29,35)(32,39,33,38);(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19)(26,32)(27,35)(28,29)(33,34)(36,37)(38,39);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165554);