#8957593

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

4574 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)(3,5)(4,7)(6,8)(10,11)(15,17)(16,18)(22,23)(25,26,31,28)(27,32,34,30)(29,33,37,36)(35,39,40,38);(1,2,6,8)(3,5,7,4)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27,31,34)(26,32,28,30)(29,35,37,40)(33,39,36,38);(1,3,6,7)(2,4,8,5)(12,19)(13,21)(15,17)(16,18)(27,34)(29,37)(30,32)(33,36);(1,4,6,5)(2,3,8,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,26)(27,30,34,32)(29,33,37,36)(35,39,40,38);(1,2)(3,4)(5,7)(6,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(1,5,6,4)(2,7,8,3)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8864789);