#8097289

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×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 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:

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

Polycyclic presentation:

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

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, 8004485);