#10504476

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

4337 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10411672);