#8215013

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	#39454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,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, 8122209);