#9015377

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

4088 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)(5,8)(6,11)(7,14)(10,15)(13,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19,23,31)(18,24,22,30)(20,28,29,21)(25,27,26,32)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,20,23,29)(18,25,22,26)(19,21,31,28)(24,32,30,27)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,20)(18,26)(19,28)(21,31)(22,25)(23,29)(24,32)(27,30)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,21,23,28)(18,27,22,32)(19,29,31,20)(24,25,30,26)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,22,23,18)(19,30,31,24)(20,25,29,26)(21,32,28,27)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,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, 8922573);