#8095042

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 8002238);