#8730723

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

4707 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8637919);