#364057

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, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=gi,h^2=i,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, 271253);