#8924383

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

4367 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831579);