#117309

order	512
nilpotency class	3
isoclinism	isoclinic class of #117305
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

567 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:

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

Polycyclic presentation:

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

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, 24505);