#8095061

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4533 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002257);