#8621832

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

4111 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529028);