#9718531

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	#49794	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,8,10,4)(5,13,15,9)(11,12,16,14)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34)(35,41)(36,40)(37,42)(38,39)(43,45)(44,46)(47,48);(2,7)(4,8)(9,13)(12,14)(20,27)(24,30)(26,31)(28,32)(33,34)(35,37)(36,44)(38,39)(40,46)(41,42)(43,45)(47,48);(2,7)(3,10)(9,13)(11,16)(18,23)(19,25)(24,30)(26,31)(34,39)(35,42)(40,44)(43,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,14)(12,13)(15,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,47)(44,45)(46,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36)(34,40)(35,43)(37,47)(38,46)(39,44)(41,45)(42,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,37)(34,35)(36,45)(38,41)(39,42)(40,48)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9625727);