#10040534

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	#45738	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,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, 9947730);