#392678

order	512
nilpotency class	3
isoclinism	isoclinic class of #195963
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3829 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 3829 conjugacy classes of subgroups — too many to draw. Only its 426 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299874);