#8750928

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,39,43)(35,42,37,40)(36,41,47,38)(44,48,46,45);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,20)(18,24,27,30)(21,28,26,31)(22,29,25,32)(33,35,39,37)(34,40,43,42)(36,46,47,44)(38,45,41,48);(2,3)(7,9)(8,10)(15,16)(19,20)(24,30)(28,31)(29,32)(33,35,39,37)(34,40,43,42)(36,46,47,44)(38,45,41,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,19)(18,24,27,30)(21,31,26,28)(22,29,25,32)(33,36,39,47)(34,41,43,38)(35,44,37,46)(40,45,42,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,25)(19,28)(20,31)(22,27)(23,26)(24,32)(29,30)(33,37,39,35)(34,42,43,40)(36,46,47,44)(38,45,41,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,25)(18,26,27,21)(19,29,20,32)(24,31,30,28)(33,38,39,41)(34,36,43,47)(35,45,37,48)(40,46,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=gi,e^2=h,f^2=gi,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, 8658124);