#8345823

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

4484 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gi,e^2=gi,f^2=gi,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, 8253019);