#10216692

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4207 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)(4,6)(5,8)(10,14)(15,17)(16,22)(23,24)(25,26)(27,31)(28,32)(29,38)(30,33)(34,36)(35,39)(37,40);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27,30,37)(26,31,33,40)(28,35,38,34)(29,36,32,39);(1,2)(3,5)(4,6)(7,8)(25,28,30,38)(26,29,33,32)(27,34,37,35)(31,39,40,36);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,28)(26,32)(27,35)(29,33)(30,38)(31,39)(34,37)(36,40);(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,27)(26,31)(28,34)(29,39)(30,37)(32,36)(33,40)(35,38);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29,30,32)(26,28,33,38)(27,36,37,39)(31,35,40,34)

Polycyclic presentation:

2^9|b^2=i,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, 10123888);