#290609

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

513 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=efi,c^2=h,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197805);