#10325748

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

3942 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:

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

Polycyclic presentation:

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

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, 10232944);