#9030050

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=h,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, 8937246);