#10069224

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

4095 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(19,25)(23,26)(27,31)(30,32)(33,34)(35,36)(37,41)(38,40)(39,42)(43,44)(45,48)(46,47);(3,9)(7,13)(10,15)(14,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35,39,38)(34,40,42,36)(37,43,45,46)(41,47,48,44);(2,6)(7,13)(8,11)(14,16)(18,20)(23,26)(24,28)(30,32)(37,45)(41,48)(43,46)(44,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36,39,40)(34,35,42,38)(37,44,45,47)(41,43,48,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,35)(34,40,42,36)(37,43,45,46)(41,44,48,47)

Polycyclic presentation:

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

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, 9976420);