#8741713

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	#54351	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8648909);