#8270481

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=ghi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8177677);