#114743

order	512
nilpotency class	3
isoclinism	isoclinic class of #114722
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

988 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21939);