#8755090

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	#53716	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,7)(5,8)(9,12)(13,15)(18,23)(19,26)(25,28)(27,32)(36,44)(39,46)(42,48)(45,47);(2,7)(3,10)(9,12)(11,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662286);