#8912130

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=ghi,e^2=g,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, 8819326);