#296781

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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

722 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)(9,10,13,17)(11,18,14,15)(12,20,21,23)(16,19,24,22)(25,26,30,35)(27,33,38,32)(28,39,37,40)(29,36,31,34);(2,5)(6,7)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,23,24,20)(25,27,30,38)(26,32,35,33)(28,29,37,31)(34,40,36,39);(9,12,14,22,13,21,11,19)(10,16,18,20,17,24,15,23)(25,28,31,27,30,37,29,38)(26,33,36,39,35,32,34,40);(1,3,4,8)(2,6,5,7)(10,17)(15,18)(16,24)(20,23)(25,29,30,31)(26,34,35,36)(27,37,38,28)(32,40,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203977);