#111031

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

575 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18227);