#10201966

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4266 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)(3,4)(5,8)(6,7)(9,10,14,15)(11,13,17,23)(12,16,21,24)(18,22,19,20)(26,31)(27,34)(33,40)(35,36);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,19)(14,17)(15,23)(16,20)(18,21)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,39)(38,40);(1,4)(2,7)(3,8)(5,6)(10,15)(11,17)(16,24)(18,19);(2,8)(4,6)(9,11)(10,13)(12,19)(14,17)(15,23)(16,20)(18,21)(22,24)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,39,40,37);(1,4)(2,3)(5,6)(7,8)(9,12,14,21)(10,16,15,24)(11,18,17,19)(13,22,23,20)(25,28,30,38)(26,33,31,40)(27,35,34,36)(29,39,32,37);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,20)(14,23)(15,17)(16,19)(18,24)(21,22)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,35,40,36)

Polycyclic presentation:

2^9|a^2=i,d^2=ghi,e^2=gh,f^2=ghi,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, 10109162);