#114232

order	512
nilpotency class	3
isoclinism	isoclinic class of #114229
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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

801 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)(3,10)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(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);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(17,19,23,33)(18,24,26,29)(20,35,38,46)(21,32,22,31)(25,42,43,48)(27,41,28,40)(30,36,47,37)(34,45,44,39);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(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,b^2=h,c^2=h,d^2=i,e^2=hi,f^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=ehi,f^c=fh

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, 21428);