#8747976

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	#45018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 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)(9,10,13,16)(11,17,18,14)(12,15,22,21)(19,24,20,23)(25,26)(27,35)(28,29)(30,33)(31,32)(34,39)(36,37)(38,40);(3,7)(5,8)(11,18)(14,17)(19,20)(23,24)(26,32)(28,34)(33,35)(36,38);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,14,16,17)(12,19,22,20)(15,23,21,24)(25,27,31,30)(26,33,32,35)(28,38,34,36)(29,37,39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,20)(13,18)(15,24)(16,17)(19,22)(21,23)(25,28,31,34)(26,29,32,39)(27,36,30,38)(33,40,35,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,21)(13,16)(15,22)(17,18)(19,24)(20,23)(25,29)(26,34)(27,37)(28,32)(30,40)(31,39)(33,36)(35,38);(9,12,13,22)(10,15,16,21)(11,19,18,20)(14,23,17,24)(25,30,31,27)(26,33,32,35)(28,36,34,38)(29,37,39,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8655172);