#302812

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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

616 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)(11,16,21,23)(12,14,17,13)(19,22,20,24)(25,26)(27,36)(28,37)(29,35)(30,34)(31,33)(32,38)(39,40);(1,3,5,4)(2,6,8,7)(9,11,15,21)(10,16,18,23)(12,22,17,24)(13,19,14,20)(25,27)(26,32)(28,33)(29,37)(30,38)(31,40)(34,36)(35,39);(9,12,14,18,15,17,13,10)(11,16,20,22,21,23,19,24)(25,28,31,36,30,39,29,32)(26,27,35,37,34,38,33,40);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,12,18,17)(11,19,21,20)(16,22,23,24)(25,29,30,31)(26,33,34,35)(27,37,38,40)(28,36,39,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=f,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210008);