#297206

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

654 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)(9,10,15,18)(11,16,21,23)(12,14,17,13)(19,22,20,24)(25,26,30,34)(27,35,38,32)(28,31,33,29)(36,40,39,37);(2,5)(6,7)(9,11,15,21)(10,16,18,23)(12,22,17,24)(13,19,14,20)(25,27,30,38)(26,32,34,35)(28,39,33,36)(29,37,31,40);(9,12,14,18,15,17,13,10)(11,16,20,22,21,23,19,24)(25,28,31,34,30,33,29,26)(27,36,37,35,38,39,40,32);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,12,18,17)(11,19,21,20)(16,22,23,24)(25,29,30,31)(26,33,34,28)(27,37,38,40)(32,36,35,39)

Polycyclic presentation:

2^9|a^2=ehi,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, 204402);