#9017775

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	#44260	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,24)(19,23,27,30)(20,25,29,32)(21,28,31,26)(33,34,40,39)(35,43,37,41)(36,42,46,48)(38,44,45,47);(2,6)(7,9)(8,12)(14,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,29)(31,32)(33,35,40,37)(34,41,39,43)(36,45,46,38)(42,44,48,47);(2,6)(3,10)(4,13)(14,15)(18,24)(19,27)(20,29)(21,31)(33,36,40,46)(34,42,39,48)(35,45,37,38)(41,44,43,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,29)(31,32)(33,37,40,35)(34,43,39,41)(36,45,46,38)(42,44,48,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,32)(26,27)(33,38,40,45)(34,44,39,47)(35,36,37,46)(41,48,43,42);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,31)(18,26,24,28)(19,25,27,32)(20,30,29,23)(33,39,40,34)(35,43,37,41)(36,42,46,48)(38,47,45,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924971);