#10055754

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=g,e^2=gi,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, 9962950);