#321025

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

632 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,24)(20,23)(25,26,31,36,30,35,29,34)(27,37,38,33,28,32,39,40);(2,5)(6,7)(9,11,17,18,13,20,12,19)(10,14,21,22,15,23,16,24)(25,27,31,38,30,28,29,39)(26,32,36,40,35,37,34,33);(11,18)(12,17)(14,22)(16,21)(19,20)(23,24)(25,28,30,27)(26,33,35,40)(29,39,31,38)(32,36,37,34);(1,3,4,8)(2,6,5,7)(10,15)(14,23)(16,21)(22,24)(25,29,30,31)(26,34,35,36)(27,38,28,39)(32,40,37,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228221);