#9003051

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	#54946	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=gh,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, 8910247);