#113872

order	512
nilpotency class	3
isoclinism	isoclinic class of #113863
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

793 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,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=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, 21068);