#8656251

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

3927 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gi,e^2=i,f^2=hi,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, 8563447);