#8581505

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488701);