#219428

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126624);