#9059051

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8966247);