#9015375

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

4272 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32)(33,34)(35,40)(36,42)(37,43)(38,39)(41,45)(44,48)(46,47);(2,6)(7,9)(8,12)(14,15)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(33,35,39,47)(34,40,38,46)(36,44,45,37)(41,43,42,48);(2,6)(3,10)(4,13)(14,15)(20,31)(21,30)(22,28)(25,32)(33,36,39,45)(34,41,38,42)(35,37,47,44)(40,48,46,43);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,36)(34,42)(35,44)(37,47)(38,41)(39,45)(40,48)(43,46);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,37,39,44)(34,43,38,48)(35,45,47,36)(40,41,46,42);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(33,38,39,34)(35,46,47,40)(36,41,45,42)(37,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=ghi,e^2=i,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, 8922571);