#352792

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dhi

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, 259988);