#8107306

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

4473 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)(3,8,4,7)(9,10)(11,12)(13,21)(14,15)(16,18)(17,22)(19,24)(20,23)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,6)(7,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,23)(21,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(2,6)(7,8)(10,15)(11,16)(17,21)(19,23)(28,37)(32,36)(34,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014502);