#8417331

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	#52653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(3,8)(4,6)(5,13)(7,9)(10,14)(11,16)(12,15)(17,18)(19,25)(20,31)(21,27)(22,32)(23,28)(24,29)(26,30)(34,36)(40,43)(41,42)(47,48);(1,3)(2,8)(4,13)(5,6)(7,12)(9,15)(10,16)(11,14)(17,19,24,22)(18,25,29,32)(20,23,30,27)(21,31,28,26)(33,34,39,36)(35,43,46,40)(37,41,38,42)(44,48,45,47);(2,9)(3,12)(6,10)(11,13)(17,20)(18,26)(19,23)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,39,46)(34,40,36,43)(37,44,38,45)(41,47,42,48);(1,4)(2,10)(3,13)(5,15)(6,9)(7,14)(8,16)(11,12)(17,21,24,28)(18,27,29,23)(19,26,22,31)(20,25,30,32)(33,36,39,34)(35,43,46,40)(37,42,38,41)(44,48,45,47);(1,5)(2,11)(3,10)(4,8)(6,12)(7,16)(9,13)(14,15)(17,22,24,19)(18,25,29,32)(20,27,30,23)(21,31,28,26)(33,37,39,38)(34,41,36,42)(35,44,46,45)(40,47,43,48);(1,6,7,10)(2,4,9,14)(3,5,12,16)(8,13,15,11)(17,23,24,27)(18,28,29,21)(19,30,22,20)(25,26,32,31)(33,38,39,37)(34,42,36,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324527);