#242084

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

709 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)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,23)(22,24)(25,26,29,32)(27,34,35,28)(30,39,38,33)(31,40,36,37);(2,5)(6,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,22,17,23)(25,27,29,35)(26,28,32,34)(30,31,38,36)(33,40,39,37);(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,28,29,34)(26,31,32,36)(27,33,35,39)(30,40,38,37);(1,3,4,7)(2,6,5,8)(11,19)(14,18)(20,24)(22,23)(27,35)(28,34)(31,36)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di,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, 149280);