#8295610

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

4086 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:

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

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=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202806);