#304693

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
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

731 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)(11,18)(12,16)(13,22)(14,17)(15,19)(20,23)(21,24)(25,26,29,33)(27,36,37,32)(28,39,38,40)(30,35,31,34);(1,3,5,4)(2,6,8,7)(25,27,29,37)(26,32,33,36)(28,30,38,31)(34,40,35,39);(1,3,5,4)(2,7,8,6)(9,11,13,19,14,20,12,21)(10,15,16,18,17,24,22,23)(25,28,31,27,29,38,30,37)(26,32,35,40,33,36,34,39);(1,4,5,3)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)(26,33)(32,36)(34,35)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,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, 211889);