#10045511

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	#48005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=gi,e^2=g,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, 9952707);