#8930689

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

4577 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:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8837885);