#330621

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
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×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

487 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237817);