#115549

order	512
nilpotency class	3
isoclinism	isoclinic class of #115542
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

913 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:

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

Polycyclic presentation:

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

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, 22745);