#10216712

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 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)(3,7)(4,5)(6,8)(10,15)(11,18)(17,24)(19,20)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,37,39,40);(2,5)(6,7)(9,10,14,15)(11,16,18,13)(12,17,22,24)(19,21,20,23)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,2)(3,6)(4,5)(7,8)(25,27,30,38)(26,32,35,31)(28,36,34,39)(29,37,33,40);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,15)(14,18)(17,23)(19,22)(21,24)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,28)(26,33)(27,36)(29,35)(30,34)(31,37)(32,40)(38,39);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,20,24,19)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=hi,e^2=ghi,f^2=gi,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, 10123908);