#8340167

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(35,37)(39,42)(43,46)(47,48);(2,7)(3,12)(9,14)(10,13)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,34)(35,42)(36,44)(37,39)(38,41)(40,45)(43,47)(46,48);(2,7)(5,16)(8,11)(9,14)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35,38,37)(34,39,41,42)(36,43,45,46)(40,47,44,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,44)(42,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,35)(34,39,41,42)(36,43,45,46)(40,48,44,47);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247363);