#10143434

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	#42557	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(18,23)(19,26)(25,32)(27,28)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,18,22,23)(19,24,26,21)(20,25,30,32)(27,29,28,31)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,4)(7,8)(9,12)(15,16)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4,6,2)(3,8,11,7)(5,12,14,9)(10,16,13,15)(17,19)(18,24)(20,28)(21,23)(22,26)(25,31)(27,30)(29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,28,32,27)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

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

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, 10050630);