#10183084

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

4648 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)(3,8,4,6)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26)(27,31)(28,29)(30,32)(33,37)(34,35)(36,38)(39,40);(3,4)(6,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,30,36)(26,31,32,38)(28,34,33,39)(29,35,37,40);(1,3,5,4)(2,6,7,8)(10,11)(15,17)(16,18)(22,23)(26,32)(28,33)(31,38)(34,39);(3,4)(6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,30,33)(26,29,32,37)(27,34,36,39)(31,35,38,40);(1,4,5,3)(2,6,7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,33)(27,35)(28,32)(30,37)(31,39)(34,38)(36,40)

Polycyclic presentation:

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

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, 10090280);