#8393463

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	#49724	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 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)(3,7,10,4)(5,13,16,9)(11,12,14,15)(34,37)(35,36)(42,45)(43,44);(1,3)(2,7)(4,8)(5,14)(6,10)(9,12)(11,16)(13,15)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,8)(3,10)(9,13)(11,14)(18,23)(21,29)(24,25)(27,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,4,6,7)(2,3,8,10)(5,15,16,12)(9,14,13,11)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,5,6,16)(2,9,8,13)(3,11,10,14)(4,12,7,15)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,3)(2,7)(4,8)(5,11)(6,10)(9,15)(12,13)(14,16)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300659);