#8517510

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

4092 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=g,f^2=g,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, 8424706);