#8739444

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	#53610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8646640);