#120311

order	512
nilpotency class	3
isoclinism	isoclinic class of #120311
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

492 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 130 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)(11,21)(12,25,28,35)(13,19)(14,18)(15,20)(16,23)(17,27,36,30)(22,31,38,40)(24,32)(26,34,37,39)(29,33);(3,8)(5,6)(9,11,13,23,15,24,14,29)(10,16,18,21,20,33,19,32)(12,26,27,22,28,37,30,38)(17,34,35,40,36,39,25,31);(2,5)(3,8)(6,7)(9,12,13,27,15,28,14,30)(10,17,18,35,20,36,19,25)(11,22,23,37,24,38,29,26)(16,31,21,34,33,40,32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27507);