#8838400

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

4339 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8745596);