#364055

order	512
nilpotency class	3
isoclinism	isoclinic class of #190432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=gi,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271251);