#10247275

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

4000 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18,23,21)(19,24,29,30)(20,26,31,25)(22,28,32,27)(33,34,39,40)(35,41,37,43)(36,46,47,42)(38,44,48,45);(2,3)(4,11)(8,10)(12,13)(17,19,23,29)(18,24,21,30)(20,27,31,28)(22,25,32,26)(34,40)(36,47)(41,43)(44,45);(2,3)(7,9)(8,10)(15,16)(17,20,23,31)(18,25,21,26)(19,28,29,27)(22,30,32,24)(33,35,39,37)(34,41,40,43)(36,45,47,44)(38,42,48,46);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,20)(18,26)(19,28)(21,25)(22,30)(23,31)(24,32)(27,29)(33,36)(34,42)(35,45)(37,44)(38,43)(39,47)(40,46)(41,48);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,21,23,18)(19,24,29,30)(20,26,31,25)(22,27,32,28)(33,37,39,35)(34,43,40,41)(36,45,47,44)(38,42,48,46);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,22)(18,27)(19,25)(20,30)(21,28)(23,32)(24,31)(26,29)(33,38,39,48)(34,44,40,45)(35,42,37,46)(36,43,47,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10154471);