#8695304

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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)(3,6)(4,7)(5,8)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,19)(10,15,13,17)(12,18,21,24)(16,22,20,23)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,31,29)(26,33,28,32)(30,35,38,37)(34,40,36,39);(1,2,4,7)(3,6,8,5)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,29,31,27)(26,32,28,33)(30,37,38,35)(34,39,36,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=h,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, 8602500);