#8274068

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

4455 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8181264);