#8122977

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
prime radical	#54858	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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)(11,12)(13,20)(14,22)(15,16)(17,23)(18,24)(19,21)(26,29)(28,36)(31,34)(33,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,12,16,20)(14,18,23,21)(17,19,22,24)(25,26,30,29)(27,31,35,34)(28,32,36,37)(33,38,39,40);(2,5)(6,8)(10,16)(11,13)(17,22)(18,21)(25,27,30,35)(26,31,29,34)(28,33,36,39)(32,38,37,40);(9,12)(10,11)(13,16)(14,19)(15,20)(17,18)(21,22)(23,24)(25,28)(26,32)(27,33)(29,37)(30,36)(31,38)(34,40)(35,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,12,16,20)(14,21,23,18)(17,19,22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,14)(10,17)(11,18)(12,19)(13,21)(15,23)(16,22)(20,24)(25,29,30,26)(27,34,35,31)(28,37,36,32)(33,40,39,38)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=h,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, 8030173);