#8581771

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

4272 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=gi,e^2=ghi,f^2=hi,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, 8488967);