#113838

order	512
nilpotency class	3
isoclinism	isoclinic class of #113838
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	#60946	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

853 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)(3,10)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(17,18)(19,28,32,24)(20,33,37,44)(21,27)(22,38)(23,26)(25,35,42,36)(29,46)(30,43,31,40)(34,45)(39,47)(41,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,15)(9,16)(10,14)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,41)(26,28)(27,43)(29,36)(33,39)(35,48)(37,47)(38,40)(42,46)(44,45);(3,11)(4,5)(7,15)(8,14)(10,16)(12,13)(17,20)(18,25)(19,29)(21,36)(22,35)(23,37)(24,39)(26,42)(27,44)(28,45)(30,34)(31,47)(32,48)(33,38)(40,46)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21034);