#8123012

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

4345 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)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(25,26,30,34)(27,33,29,31)(28,32,37,40)(35,39,36,38);(1,2)(3,6)(4,5)(7,8)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(25,27,30,29)(26,31,34,33)(28,35,37,36)(32,38,40,39);(2,5)(6,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,28,30,37)(26,32,34,40)(27,35,29,36)(31,38,33,39);(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,3,4,7)(2,6,5,8)(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,29,30,27)(26,33,34,31)(28,36,37,35)(32,39,40,38)

Polycyclic presentation:

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

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, 8030208);