#8734400

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

4159 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)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18,23,24)(19,21,20,27)(22,25,31,32)(26,29,28,30)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(2,6)(7,9)(8,12)(15,16)(17,19)(18,21)(20,23)(22,26)(24,27)(25,29)(28,31)(30,32)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(2,6)(7,9)(8,12)(15,16)(18,24)(19,20)(25,32)(26,28)(36,44)(41,46)(43,45)(47,48);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,20,23,19)(18,21,24,27)(22,26,31,28)(25,30,32,29)(33,36,39,44)(34,41,38,46)(35,43,37,45)(40,47,42,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,19)(20,24)(22,30)(23,27)(25,28)(26,32)(29,31)(33,37,39,35)(34,40,38,42)(36,45,44,43)(41,47,46,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,32)(27,30)(33,38,39,34)(35,42,37,40)(36,41,44,46)(43,47,45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8641596);