#8097840

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8005036);