#8755339

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

3983 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,20)(21,29)(24,27)(26,30)(34,39)(37,45)(40,46)(44,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,23,20)(19,24,22,27)(21,28,29,25)(26,32,30,31)(36,43)(40,46)(42,47)(44,48);(2,6)(7,9)(8,12)(14,15)(17,19,23,22)(18,24,20,27)(21,26,29,30)(25,31,28,32)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,18)(19,24,22,27)(21,28,29,25)(26,31,30,32)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,36,38,43)(34,40,39,46)(35,42,41,47)(37,44,45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22,23,19)(18,24,20,27)(21,30,29,26)(25,31,28,32)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662535);