#382449

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2606 conjugacy classes of subgroups; only the 418 normal subgroups are shown.

normal subgroups only

This group has 2606 conjugacy classes of subgroups — too many to draw. Only its 418 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289645);