#8595759

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
prime radical	#40783	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4605 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8502955);