#8109975

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	#37198	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4456 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)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,31,29)(27,32,37,35)(28,38,36,33)(30,39,40,34);(2,3)(6,7)(10,14)(12,21)(15,17)(18,24)(26,29)(32,35)(33,38)(34,39);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,34,38,39);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,36)(26,33,29,38)(27,30,37,40)(32,34,35,39);(1,3,5,2)(4,7,8,6)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29,31,26)(27,35,37,32)(28,38,36,33)(30,39,40,34);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017171);