#113627

order	512
nilpotency class	3
isoclinism	isoclinic class of #113623
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	#60967	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 20823);