#393784

order	512
nilpotency class	3
isoclinism	isoclinic class of #195963
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

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×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1848 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 1848 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,d^2=g,h^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, 300980);