#304894

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

696 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)(11,13)(12,19)(14,18)(15,21)(16,22)(17,23)(20,24)(25,26)(27,33)(28,39)(29,36)(30,35)(31,34)(32,37)(38,40);(1,3,5,4)(2,6,8,7)(9,11,13,10)(12,17,20,15)(14,18,22,16)(19,21,24,23)(25,27,30,37)(26,32,35,33)(28,29,38,31)(34,40,36,39);(1,3,5,4)(2,7,8,6)(9,12,14,21,13,20,22,23)(10,15,16,19,11,17,18,24)(25,28,31,27,30,38,29,37)(26,33,36,39,35,32,34,40);(1,4,5,3)(2,7,8,6)(10,11)(12,20)(16,18)(21,23)(25,29,30,31)(26,34,35,36)(27,28,37,38)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=efh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

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, 212090);