#110927

order	512
nilpotency class	3
isoclinism	isoclinic class of #110922
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

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

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, 18123);