#8634092

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	#35682	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,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, 8541288);