#8122914

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

4351 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8030110);