#9040406

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	#49681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=g,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, 8947602);