#10042543

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(5,13)(6,7)(10,14)(11,15)(12,16)(17,18,23,25)(19,24,20,30)(21,26,27,22)(28,29,31,32)(33,34)(35,44)(36,37)(38,48)(39,41)(40,45)(42,47)(43,46);(2,7)(3,10)(4,13)(12,15)(17,19)(18,24)(20,23)(21,31)(22,29)(25,30)(26,32)(27,28)(33,35,39,45)(34,40,41,44)(36,46,47,38)(37,48,42,43);(2,7)(3,10)(4,13)(12,15)(18,25)(19,20)(21,27)(29,32)(34,41)(35,45)(36,47)(43,48);(1,3,6,10)(2,8,7,14)(4,11,13,16)(5,12,9,15)(17,20,23,19)(18,24,25,30)(21,31,27,28)(22,32,26,29)(33,36)(34,42)(35,46)(37,41)(38,45)(39,47)(40,43)(44,48);(1,4,6,13)(2,9,7,5)(3,11,10,16)(8,15,14,12)(17,21)(18,26)(19,28)(20,31)(22,25)(23,27)(24,29)(30,32)(33,37,39,42)(34,36,41,47)(35,43,45,48)(38,44,46,40);(1,5,6,9)(2,4,7,13)(3,12,10,15)(8,11,14,16)(17,22,23,26)(18,27,25,21)(19,29,20,32)(24,28,30,31)(33,38,39,46)(34,43,41,48)(35,36,45,47)(37,44,42,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gi,e^2=g,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, 9949739);