#9027797

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	#54176	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3919 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,31,28,29)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(2,6)(3,4)(5,13)(15,16)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,29,32,31)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)(33,38,39,44)(34,42,43,48)(35,45,46,36)(37,47,40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934993);