#8607447

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
prime radical	#46768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,29)(27,31)(28,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(34,35)(39,41)(40,42)(46,47);(2,5)(8,10)(9,12)(15,16)(18,23)(20,30)(24,26)(27,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,28)(30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,31)(29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,32,26,27)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8514643);