#111983

order	512
nilpotency class	3
isoclinism	isoclinic class of #111981
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#37606	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,g^c=gh

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, 19179);