#110212

order	512
nilpotency class	3
isoclinism	isoclinic class of #110189
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

595 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17408);