#113831

order	512
nilpotency class	3
isoclinism	isoclinic class of #113829
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	#60953	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

759 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:

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

Polycyclic presentation:

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

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, 21027);