#9040310

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^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, 8947506);