#10506274

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10413470);