#115551

order	512
nilpotency class	3
isoclinism	isoclinic class of #115542
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1057 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)(3,7)(4,6)(5,8)(9,10,14,19,15,18,13,28)(11,16,21,33,22,31,29,32)(12,23,27,36,25,17,26,35)(20,37,24,34,38,30,39,40);(2,6)(3,4)(7,8)(9,11)(10,16)(12,24)(13,29)(14,21)(15,22)(17,34)(18,31)(19,33)(20,26)(23,40)(25,39)(27,38)(28,32)(30,35)(36,37);(1,3,5,4)(2,7,8,6)(9,12)(10,17)(11,20)(13,27)(14,26)(15,25)(16,30)(18,23)(19,36)(21,39)(22,38)(24,29)(28,35)(31,37)(32,40)(33,34)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,f^2=i,g^2=i,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, 22747);