#8930274

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

4663 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,12,17,16)(13,21,22,18)(19,23,24,20)(27,33)(31,38)(34,36)(39,40);(1,2,6,8)(3,5,7,4)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,23)(22,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,3,6,7)(2,4,8,5)(10,15)(11,17)(13,22)(20,23)(26,30)(27,33)(32,35)(34,36);(1,4,6,5)(2,3,8,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,27,29,33)(26,31,30,38)(28,36,37,34)(32,40,35,39);(1,2)(3,4)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(1,5,6,4)(2,7,8,3)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=gh,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, 8837470);