#10040978

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

4029 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gi,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, 9948174);