#8406831

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

4050 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=g,f^2=g,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, 8314027);