#9814831

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

4200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=gi,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, 9722027);