#10040543

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	#56409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,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, 9947739);