#8908002

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

4654 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=gh,f^2=h,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, 8815198);