#8549548

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,e^2=gh,f^2=hi,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, 8456744);