#8751771

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	#47658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658967);