#8720813

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	#56381	C₂
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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,30)(21,32)(23,27)(24,29)(25,31)(26,28)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,13)(15,16)(17,19)(18,24)(20,21)(22,27)(23,29)(25,32)(26,30)(28,31)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(2,7)(3,8)(9,16)(11,13)(17,20,23,31)(18,25,27,30)(19,21,29,28)(22,26,24,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,3,6,8)(2,4,7,10)(5,13,15,11)(9,14,16,12)(17,20)(18,25)(19,28)(21,29)(22,32)(23,31)(24,26)(27,30)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,23,28)(18,26,27,32)(19,20,29,31)(22,25,24,30)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,5,6,15)(2,9,7,16)(3,11,8,13)(4,12,10,14)(17,22,23,24)(18,19,27,29)(20,26,31,32)(21,30,28,25)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8628009);