#8159536

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

4147 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)(3,5)(4,9)(6,14)(7,8)(10,15)(11,16)(12,13)(17,18,22,24)(19,25,20,23)(21,29,30,26)(27,31,28,32)(37,45)(40,46)(42,47)(43,48);(1,3)(2,5)(4,7)(6,11)(8,9)(10,13)(12,15)(14,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,46)(47,48);(2,8)(3,4)(10,14)(11,12)(18,24)(23,25)(26,29)(31,32)(34,39)(35,41)(40,46)(42,47);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,35,38,41)(34,36,39,44)(37,42,45,47)(40,43,46,48);(1,5,7,9)(2,4,8,3)(6,13,15,16)(10,12,14,11)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,30)(18,26,24,29)(19,27,20,28)(23,31,25,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066732);