#8402374

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	#34962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gi,f^2=gi,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, 8309570);