#8127406

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

4200 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034602);