#8824347

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	#48538	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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)(4,10)(5,6)(7,9)(11,15)(12,16)(13,14)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3)(2,8)(4,11)(5,7)(6,9)(10,15)(12,13)(14,16)(19,25)(21,29)(23,26)(24,28)(34,38)(36,44)(39,41)(42,48);(2,9)(3,5)(4,12)(14,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,7,12)(2,10,9,16)(3,11,5,13)(6,14,8,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,7,3)(2,6,9,8)(4,13,12,11)(10,14,16,15)(17,20,22,18)(19,23,25,26)(21,28,29,24)(27,30,31,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,6)(2,3)(4,14)(5,9)(7,8)(10,11)(12,15)(13,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731543);