#120328

order	512
nilpotency class	3
isoclinism	isoclinic class of #10530
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

488 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,f^b=fh,e^c=ehi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27524);