#115555

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

921 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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=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, 22751);