#10457955

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

4711 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)(6,8)(9,10,14,17)(11,18,12,15)(13,21,22,16)(19,23,20,24)(25,26)(27,30)(28,34)(29,32)(31,35)(33,39)(36,40)(37,38);(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27,31,29)(26,32,35,30)(28,36,39,37)(33,40,34,38);(1,2,5,4)(3,6,7,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,28)(26,33)(27,36)(29,37)(30,38)(31,39)(32,40)(34,35);(3,7)(6,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,28)(26,34)(27,37)(29,36)(30,38)(31,39)(32,40)(33,35);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,29,31,27)(26,32,35,30)(28,37,39,36)(33,40,34,38);(1,4,5,2)(3,8,7,6)(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(25,30,31,32)(26,29,35,27)(28,38,39,40)(33,37,34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=g,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, 10365151);