#10312675

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

4202 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10219871);