#119445

order	512
nilpotency class	3
isoclinism	isoclinic class of #119443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

684 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 26641);