#8393638

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

4130 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:

(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18,22,23)(19,27,30,24)(20,21,26,25)(28,29,31,32);(2,6)(3,9)(8,13)(11,15)(18,23)(20,26)(24,27)(28,31)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(4,12)(8,13)(11,15)(14,16)(18,23)(21,25)(24,27)(29,32)(34,39)(37,45)(40,41)(43,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,30)(18,24,23,27)(20,31,26,28)(21,29,25,32)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,22,26)(18,25,23,21)(19,28,30,31)(24,29,27,32)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,25)(18,26,23,20)(19,29,30,32)(24,28,27,31)(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=i,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, 8300834);