#10042520

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9949716);