#8908065

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

4093 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)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,25)(20,28)(21,27)(22,23)(24,29)(26,31)(30,32)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,47,44);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,19,22,27)(18,21,23,25)(20,26,29,30)(24,31,28,32)(33,35)(34,39)(36,44)(37,40)(38,43)(41,45)(42,46)(47,48);(2,6)(3,4)(10,13)(11,12)(18,23)(20,29)(21,25)(26,30)(35,43)(36,45)(39,46)(40,48);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,32)(27,30)(33,36)(34,40)(35,44)(37,39)(38,45)(41,43)(42,48)(46,47);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,19)(18,25)(20,30)(21,23)(22,27)(24,31)(26,29)(28,32)(33,37)(34,41)(35,40)(36,46)(38,47)(39,45)(42,44)(43,48);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,21,22,25)(18,19,23,27)(20,31,29,32)(24,26,28,30)(33,35)(34,39)(36,44)(37,40)(38,43)(41,45)(42,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815261);