#8301810

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

4270 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8209006);