#8120057

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	#46826	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=h,e^2=i,f^2=g,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, 8027253);