#8534858

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

4258 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:

(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19,21,27)(18,22,23,25)(20,26,29,28)(24,30,31,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(3,9)(6,13)(10,15)(14,16)(18,23)(22,25)(24,31)(30,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)(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=h,c^2=i,e^2=ghi,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, 8442054);