#10040995

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

4252 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gi,e^2=g,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, 9948191);