#110692

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

570 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17888);