#8716815

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	#43039	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4338 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8624011);