#9815092

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

4402 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9722288);