#111443

order	512
nilpotency class	3
isoclinism	isoclinic class of #111436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

776 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,9,13,16)(3,8,11,15,4,7,12,14)(17,18)(19,29)(20,34,37,25)(21,28)(22,26)(23,27)(24,31)(30,42,45,47)(32,40)(33,41)(35,39,46,48)(36,43,38,44);(1,3,6,4)(2,7,9,8)(5,12,13,11)(10,15,16,14)(17,19,23,32)(18,24,27,41)(20,35,37,46)(21,33,22,31)(25,42,34,47)(26,40,28,29)(30,38,45,36)(39,44,48,43);(1,4,6,3)(2,8,9,7)(5,11,13,12)(10,14,16,15)(17,20,23,37)(18,25,27,34)(19,30,32,45)(21,36,22,38)(24,39,41,48)(26,43,28,44)(29,42,40,47)(31,35,33,46)

Polycyclic presentation:

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

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, 18639);