#8634840

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	#45509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4400 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,e^2=gh,f^2=gi,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, 8542036);