#8664402

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

4522 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,16,20,15)(12,21,22,13)(18,19,24,23)(25,26)(27,33)(28,38)(29,34)(30,31)(32,35)(36,39)(37,40);(2,5)(6,7)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,23)(22,24)(25,27,32,30)(26,33,35,31)(28,37,29,36)(34,39,38,40);(1,2)(3,6)(4,5)(7,8)(25,28,32,29)(26,34,35,38)(27,36,30,37)(31,39,33,40);(1,3,4,8)(2,6,5,7)(9,12,14,22)(10,13,17,21)(11,18,20,24)(15,19,16,23)(25,29,32,28)(26,34,35,38)(27,37,30,36)(31,39,33,40);(9,11)(10,16)(12,18)(13,23)(14,20)(15,17)(19,21)(22,24)(25,30,32,27)(26,33,35,31)(28,37,29,36)(34,40,38,39);(9,13)(10,12)(11,19)(14,21)(15,18)(16,24)(17,22)(20,23)(25,31,32,33)(26,30,35,27)(28,39,29,40)(34,37,38,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=h,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, 8571598);