#8758019

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4468 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8665215);