#8114362

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

4236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021558);