#8757301

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

3977 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=i,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, 8664497);