#8093831

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

4566 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8001027);