#8391338

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4306 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8298534);