#8158963

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	#54190	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066159);