#8757924

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

4650 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8665120);