#9720391

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

3818 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=ghi,f^2=ghi,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, 9627587);