#8406848

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

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

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, 8314044);