#9040709

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	#50749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(3,8)(4,6)(5,10)(7,9)(11,15)(12,14)(13,16)(33,34,40,42)(35,37,39,36)(38,41,47,48)(43,45,46,44);(1,3)(2,8)(4,5)(6,10)(7,12)(9,14)(11,13)(15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35,40,39)(34,36,42,37)(38,46,47,43)(41,45,48,44);(2,9)(3,12)(5,13)(6,15)(18,22)(20,29)(23,25)(26,30)(33,36,40,37)(34,35,42,39)(38,45,47,44)(41,46,48,43);(1,4,7,11)(2,6,9,15)(3,13,12,5)(8,16,14,10)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,37,40,36)(34,35,42,39)(38,44,47,45)(41,46,48,43);(1,5,7,13)(2,10,9,16)(3,4,12,11)(6,14,15,8)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,38,40,47)(34,41,42,48)(35,43,39,46)(36,44,37,45);(1,6)(2,11)(3,10)(4,9)(5,14)(7,15)(8,13)(12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,39,40,35)(34,36,42,37)(38,46,47,43)(41,44,48,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^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, 8947905);