#10506325

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	#49798	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,e^2=hi,f^2=h,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, 10413521);