#9040508

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
prime radical	#49863	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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)(3,8)(4,6)(5,10)(7,9)(11,15)(12,16)(13,14)(33,34)(35,40)(36,45)(37,41)(38,42)(39,43)(44,48)(46,47);(1,3,7,12)(2,8,9,16)(4,14,11,5)(6,13,15,10)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,34,39,43)(35,40,36,45)(37,46,47,41)(38,42,48,44);(2,9)(3,12)(5,14)(6,15)(18,22)(20,29)(23,25)(26,30)(33,35,39,36)(34,40,43,45)(37,42,47,44)(38,46,48,41);(1,4,7,11)(2,6,9,15)(3,5,12,14)(8,10,16,13)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,36,39,35)(34,40,43,45)(37,44,47,42)(38,46,48,41);(1,5)(2,10)(3,4)(6,8)(7,14)(9,13)(11,12)(15,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37)(34,41)(35,44)(36,42)(38,45)(39,47)(40,48)(43,46);(1,6)(2,11)(3,13)(4,9)(5,8)(7,15)(10,12)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,38,39,48)(34,42,43,44)(35,41,36,46)(37,45,47,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^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=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947704);