#8159252

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	#45417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4325 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)(3,7,8,5)(9,10)(11,13)(12,20)(14,15)(16,22)(17,18)(19,24)(21,23)(27,28)(31,32)(35,36)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,26,30,34)(27,31,28,32)(29,33,37,40)(35,38,36,39);(1,3)(2,5)(4,8)(6,7)(10,15)(11,17)(16,20)(19,23)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,20)(13,17)(14,15)(16,22)(19,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,22)(10,16,15,20)(11,19,17,23)(13,21,18,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,13,14,18)(10,17,15,11)(12,21,22,24)(16,23,20,19)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066448);