#319002

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
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

520 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)(11,16)(12,21)(13,18)(14,23)(15,19)(17,22)(20,24)(25,26,28,33)(27,35,37,31)(29,32,30,34)(36,39,38,40);(1,3,5,4)(2,6,8,7)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27,29,36,28,37,30,38)(26,31,32,40,33,35,34,39);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(26,33)(31,35)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=eghi,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,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, 226198);