#8508887

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	#39256	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416083);