#8696128

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	#57996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603324);