#9267059

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

4019 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)(3,8)(4,10)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,43,46,48);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(19,25)(22,29)(26,31)(30,32)(34,39)(37,45)(40,41)(43,48);(2,9)(3,4)(5,11)(13,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35)(34,40)(36,46)(37,48)(38,44)(39,41)(42,47)(43,45);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,37)(35,42)(38,47)(39,45)(40,43)(41,48)(44,46);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37)(34,36)(35,43)(38,45)(39,47)(40,42)(41,46)(44,48);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174255);