#8360923

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

4043 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8268119);