#9717584

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	#55015	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(34,39)(35,42)(41,47)(43,44);(4,11)(8,14)(10,15)(12,16)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(2,7)(3,9)(8,14)(10,15)(18,20)(19,25)(24,28)(27,29)(33,35,38,42)(34,40,39,37)(36,44,46,43)(41,45,47,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,26,25,22)(21,28,31,24)(27,30,29,32)(33,36,38,46)(34,41,39,47)(35,43,42,44)(37,45,40,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,31)(18,24,20,28)(19,27,25,29)(22,32,26,30)(33,37)(34,35)(36,45)(38,40)(39,42)(41,43)(44,47)(46,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=i,f^2=i,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, 9624780);