#297205

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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

646 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=f,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204401);