#10534366

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	#39300	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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)(4,9)(5,10)(6,7)(11,12)(13,15)(14,16)(17,18)(19,23)(20,24)(21,27)(22,25)(26,29)(28,31)(30,32)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,47,46,45);(2,7)(3,10)(9,15)(11,16)(17,19,22,27)(18,23,25,21)(20,26,30,28)(24,29,32,31);(17,19)(18,21)(20,26)(22,27)(23,25)(24,31)(28,30)(29,32)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,12,15,14)(17,19)(18,21)(20,28)(22,27)(23,25)(24,29)(26,30)(31,32)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,20)(18,24)(19,26)(21,31)(22,30)(23,29)(25,32)(27,28)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,46)(42,44);(1,5,6,8)(2,10,7,3)(4,12,13,14)(9,11,15,16)(17,21)(18,19)(20,29)(22,23)(24,28)(25,27)(26,32)(30,31)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10441562);