#8720491

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=i,e^2=h,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, 8627687);