#10519363

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	#43188	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(34,39)(35,40)(37,45)(43,46);(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,19,23,24)(18,21,26,20)(22,27,31,32)(25,29,30,28)(34,39)(35,40)(41,47)(42,48);(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,34,38,39)(35,36,40,44)(37,41,45,47)(42,43,48,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,48)(46,47);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,28)(29,32)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,45)(47,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37,38,45)(34,41,39,47)(35,42,40,48)(36,43,44,46)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=ghi,f^2=gh,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, 10426559);