#8094662

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

4122 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(18,22)(19,20)(26,29)(27,28)(36,44)(40,45)(42,48)(46,47);(3,9)(7,13)(10,15)(14,16)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31)(34,38)(39,41)(40,45)(46,47);(2,6)(7,13)(8,11)(14,16)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,23,19)(18,25,22,24)(21,28,30,27)(26,32,29,31);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8001858);