#392117

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×2₋¹⁺⁴	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

15806 conjugacy classes of subgroups; only the 698 normal subgroups are shown.

normal subgroups only

This group has 15806 conjugacy classes of subgroups — too many to draw. Only its 698 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)(9,10,13,18,14,17,26,31)(11,19,21,32,22,15,25,28)(12,16,24,30,23,29,36,39)(20,33,35,40,34,27,37,38);(9,11,14,22)(10,15,17,19)(12,20,23,34)(13,25,26,21)(16,27,29,33)(18,32,31,28)(24,37,36,35)(30,40,39,38);(2,4)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,29)(18,30)(19,33)(21,35)(22,34)(25,37)(26,36)(28,38)(31,39)(32,40);(6,8)(12,23)(16,29)(20,34)(24,36)(27,33)(30,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=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, 299313);