#8994057

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

4512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901253);