#10040734

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	#58834	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 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)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,21)(19,25,27,20)(22,29,30,24)(26,28,32,31)(33,34)(35,40)(36,44)(37,42)(38,46)(39,43)(41,47)(45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,27)(18,20,21,25)(22,26,30,32)(24,28,29,31)(33,35,39,38)(34,40,43,46)(36,45,47,42)(37,44,48,41);(2,3)(7,9)(8,10)(14,15)(18,21)(20,25)(24,29)(28,31)(33,36,39,47)(34,41,43,44)(35,42,38,45)(37,40,48,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,36)(34,41)(35,42)(37,46)(38,45)(39,47)(40,48)(43,44);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,18)(19,20,27,25)(22,24,30,29)(26,31,32,28)(33,37)(34,42)(35,41)(36,46)(38,44)(39,48)(40,47)(43,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,30)(18,24,21,29)(19,26,27,32)(20,28,25,31)(33,38,39,35)(34,40,43,46)(36,45,47,42)(37,41,48,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,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, 9947930);