#9910071

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

4680 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817267);