#8696112

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	#44312	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(25,26)(27,30)(28,34)(29,39)(31,33)(32,35)(36,37)(38,40);(3,7)(6,8)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(25,27,32,31)(26,33,35,30)(28,36,29,38)(34,40,39,37);(1,2,5,4)(3,6,7,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,28,32,29)(26,34,35,39)(27,36,31,38)(30,37,33,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,29,32,28)(26,34,35,39)(27,36,31,38)(30,40,33,37);(1,4,5,2)(3,8,7,6)(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,30)(26,27)(28,37)(29,40)(31,35)(32,33)(34,36)(38,39);(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,31,32,27)(26,33,35,30)(28,38,29,36)(34,40,39,37)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=h,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, 8603308);