#8188502

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095698);