#113388

order	512
nilpotency class	3
isoclinism	isoclinic class of #113385
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,9,7,4,5,10)(3,11,13,8,14,12,16,15)(17,18,22,27,23,26,21,37)(19,28,31,44,32,24,30,40)(20,33,35,25,36,43,38,42)(29,46,34,45,47,41,48,39);(1,3,7,14)(2,8,4,15)(5,16,6,13)(9,12,10,11)(17,19,23,32)(18,24,26,28)(20,34,36,48)(21,31,22,30)(25,41,42,46)(27,44,37,40)(29,38,47,35)(33,39,43,45);(1,4)(2,7)(3,12)(5,10)(6,9)(8,13)(11,14)(15,16)(17,20)(18,25)(19,29)(21,38)(22,35)(23,36)(24,39)(26,42)(27,43)(28,45)(30,48)(31,34)(32,47)(33,37)(40,46)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20584);