#290230

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
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

535 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)(9,10,14,18,13,17,20,23)(11,19,21,16,12,15,22,24)(25,26,31,37,29,28,30,33)(27,34,38,39,35,32,36,40);(1,3,5,4)(2,6,8,7)(9,11,20,22,13,12,14,21)(10,15,23,16,17,19,18,24)(25,27,31,38,29,35,30,36)(26,32,37,40,28,34,33,39);(9,12,13,11)(10,16,17,24)(14,21,20,22)(15,18,19,23)(25,28,29,26)(27,34,35,32)(30,33,31,37)(36,39,38,40);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=efhi,b^2=ef,c^2=h,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197426);