#8900129

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	#51769	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8807325);