#8393432

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gi,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, 8300628);