#391946

order	512
nilpotency class	3
isoclinism	isoclinic class of #213555
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

916 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299142);