#114237

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

828 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=hi,f^2=h,g^2=i,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, 21433);