#9947480

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854676);