#8741931

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	#52324	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(2,7)(3,8)(9,16)(10,13)(18,24)(19,20)(26,31)(27,28)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,7)(3,8)(9,16)(10,13)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,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, 8649127);