#111320

order	512
nilpotency class	3
isoclinism	isoclinic class of #111318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,7,8,5)(4,6)(9,10)(11,20,23,31)(12,24)(13,18)(14,29)(15,19)(16,22,32,28)(17,26)(21,33,38,30)(25,34)(27,35)(36,39,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,33)(18,31)(19,32)(20,29)(24,40)(25,39)(26,37)(27,38)(30,34)(35,36);(1,2)(3,5)(4,6)(7,8)(9,12,15,27)(10,17,19,34)(11,21,23,38)(13,26,14,25)(16,30,32,33)(18,35,29,24)(20,36,31,40)(22,39,28,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,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, 18516);