#112135

order	512
nilpotency class	3
isoclinism	isoclinic class of #112133
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

863 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(17,18)(19,27,30,23)(20,31,33,24)(21,35)(22,25)(26,38)(28,39)(29,40,36,43)(32,48)(34,42,37,44)(41,47)(45,46);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(17,19,22,30)(18,23,25,27)(20,32,33,47)(21,36,38,29)(24,41,31,48)(26,43,35,40)(28,37,46,34)(39,44,45,42);(4,13)(6,14)(7,16)(10,12)(17,20)(18,24)(19,28)(21,37)(22,33)(23,39)(25,31)(26,44)(27,45)(29,47)(30,46)(32,36)(34,38)(35,42)(40,48)(41,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,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, 19331);