#8907013

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3860 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:

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

Polycyclic presentation:

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

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, 8814209);