#8664431

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

4563 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)(3,8)(4,5)(6,7)(9,10)(11,17)(12,22)(13,18)(14,15)(16,19)(20,23)(21,24)(25,26,30,31)(27,33,28,32)(29,34,38,37)(35,40,36,39);(1,3,6,5)(2,4,7,8)(9,11,16,14)(10,17,19,15)(12,21,13,20)(18,23,22,24)(26,31)(32,33)(34,37)(39,40);(9,12,16,13)(10,18,19,22)(11,20,14,21)(15,23,17,24)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(1,4,6,8)(2,3,7,5)(9,13,16,12)(10,18,19,22)(11,21,14,20)(15,23,17,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(1,2)(3,4)(5,8)(6,7)(9,14,16,11)(10,17,19,15)(12,21,13,20)(18,24,22,23)(25,28,30,27)(26,33,31,32)(29,35,38,36)(34,39,37,40);(1,5,6,3)(2,4,7,8)(9,15,16,17)(10,14,19,11)(12,23,13,24)(18,21,22,20)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8571627);