#10247277

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

3975 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)(3,10)(4,5)(6,8)(7,14)(9,11)(12,13)(15,16)(17,18,23,21)(19,24,29,30)(20,26,31,25)(22,28,32,27)(33,34,39,44)(35,41,37,40)(36,46,47,42)(38,43,48,45);(1,3,6,14)(2,7,8,10)(4,15,11,12)(5,13,9,16)(17,19,23,29)(18,24,21,30)(20,27,31,28)(22,25,32,26)(33,35,39,37)(34,40,44,41)(36,43,47,45)(38,42,48,46);(2,8)(4,11)(7,10)(12,15)(17,20,23,31)(18,25,21,26)(19,28,29,27)(22,30,32,24)(33,35,39,37)(34,41,44,40)(36,45,47,43)(38,42,48,46);(1,4,6,11)(2,9,8,5)(3,12,14,15)(7,13,10,16)(17,20)(18,26)(19,28)(21,25)(22,30)(23,31)(24,32)(27,29)(33,36)(34,42)(35,45)(37,43)(38,40)(39,47)(41,48)(44,46);(1,3)(2,10)(4,15)(5,16)(6,14)(7,8)(9,13)(11,12)(17,21,23,18)(19,24,29,30)(20,26,31,25)(22,27,32,28)(33,37,39,35)(34,40,44,41)(36,45,47,43)(38,42,48,46);(1,5,6,9)(2,11,8,4)(3,13,14,16)(7,12,10,15)(17,22)(18,27)(19,25)(20,30)(21,28)(23,32)(24,31)(26,29)(33,38,39,48)(34,43,44,45)(35,42,37,46)(36,40,47,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,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, 10154473);