#8321917

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229113);