#8393640

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

4115 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=hi,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, 8300836);