#112144

order	512
nilpotency class	3
isoclinism	isoclinic class of #112133
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

830 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,g^2=hi,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,f^c=fh,g^c=gh

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, 19340);