#114233

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	#60965	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

811 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:

(1,2,5,9,6,7,4,13)(3,10,11,8,12,16,14,15)(17,18)(19,28,32,43)(20,33)(21,26)(22,36)(23,27)(24,30,39,31)(25,34)(29,38,46,40)(35,41)(37,42)(44,47,48,45);(2,7)(3,11)(4,5)(8,16)(10,15)(12,14)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,40)(26,43)(27,39)(28,36)(33,48)(34,47)(35,46)(37,45)(38,42)(41,44);(1,3)(2,8)(4,14)(5,11)(6,12)(7,15)(9,16)(10,13)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,41)(27,42)(28,44)(30,45)(31,47)(32,46)(33,36)(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=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, 21429);