#10451637

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10358833);