#10106222

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	#40173	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,22,25)(19,23,29,32)(20,21,30,24)(26,31,27,28)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,43,45,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,22,29)(18,23,25,32)(20,27,30,26)(21,28,24,31)(33,35,40,46)(34,41,38,44)(36,45,37,39)(42,43,47,48);(2,7)(6,13)(8,11)(14,15)(17,20,22,30)(18,24,25,21)(19,26,29,27)(23,28,32,31)(33,36,40,37)(34,42,38,47)(35,45,46,39)(41,43,44,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,21)(19,27)(22,30)(23,28)(24,25)(26,29)(31,32)(33,37,40,36)(34,42,38,47)(35,39,46,45)(41,43,44,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,43,45,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,21)(18,20)(19,28)(22,24)(23,27)(25,30)(26,32)(29,31)(33,39)(34,43)(35,36)(37,46)(38,48)(40,45)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10013418);