#8093794

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

4316 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)(11,13)(12,17)(14,24)(15,18)(16,21)(19,23)(20,22)(25,26)(27,33)(28,35)(29,32)(30,31)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,18)(14,20)(15,21)(17,22)(23,24)(26,31)(28,33)(32,38)(36,39);(2,5)(6,8)(9,12,15,14)(10,17,18,24)(11,19,21,20)(13,23,16,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(9,11)(10,16)(12,19)(13,18)(14,20)(15,21)(17,22)(23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,16)(10,11,18,21)(12,22,14,23)(17,20,24,19);(1,3)(2,6)(4,7)(5,8)(9,14,15,12)(10,17,18,24)(11,20,21,19)(13,23,16,22)(25,29,30,37)(26,32,31,38)(27,34,35,40)(28,36,33,39)

Polycyclic presentation:

2^9|c^2=gh,e^2=gh,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, 8000990);