#8824425

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	#36332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,8)(4,5)(6,7)(9,12)(10,16)(11,13)(14,15)(18,23)(21,24)(25,30)(29,31);(2,7)(5,15)(8,9)(11,16)(20,28)(25,30)(27,32)(29,31)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(1,3,6,12)(2,8,7,9)(4,13,14,10)(5,11,15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,9)(4,10)(5,11)(6,12)(7,8)(13,14)(15,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,4)(2,5)(3,10)(6,14)(7,15)(8,16)(9,11)(12,13)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,5)(2,4)(3,11)(6,15)(7,14)(8,13)(9,10)(12,16)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731621);