#8756145

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663341);