#111972

order	512
nilpotency class	3
isoclinism	isoclinic class of #111967
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,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, 19168);