#8534545

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

3937 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441741);