#8215210

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

4313 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8122406);