#9720410

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9627606);