#8810107

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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)(3,8)(4,10)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,21)(20,24)(22,26)(23,28)(25,30)(27,31)(29,32)(34,36)(35,38)(41,44)(42,45);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,22,28)(18,23,26,21)(20,27,30,32)(24,29,25,31)(33,34,39,36)(35,40,38,43)(37,44,46,41)(42,48,45,47);(2,9)(3,4)(5,11)(13,15)(17,20,22,30)(18,24,26,25)(19,27,28,32)(21,31,23,29)(33,35,39,38)(34,40,36,43)(37,45,46,42)(41,48,44,47);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,26)(28,32)(33,36,39,34)(35,40,38,43)(37,44,46,41)(42,47,45,48);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,21)(18,19)(20,29)(22,23)(24,32)(25,27)(26,28)(30,31)(33,37,39,46)(34,41,36,44)(35,42,38,45)(40,47,43,48);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,19)(18,21)(20,27)(22,28)(23,26)(24,31)(25,29)(30,32)(33,38,39,35)(34,40,36,43)(37,45,46,42)(41,47,44,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717303);