#8097096

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 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)(11,17)(12,16)(13,21)(14,18)(15,23)(19,22)(20,24)(25,26,31,36)(27,30,35,32)(28,33,38,40)(29,39,37,34);(2,6)(4,7)(9,11)(10,13)(12,20)(14,21)(15,24)(16,22)(17,18)(19,23)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,15)(11,19)(13,22)(14,23)(16,18)(17,24)(20,21)(25,28,31,38)(26,33,36,40)(27,37,35,29)(30,34,32,39);(1,4,5,7)(2,3,6,8)(9,13,14,17)(10,11,18,21)(12,22,23,24)(15,19,16,20)(25,29,31,37)(26,34,36,39)(27,38,35,28)(30,33,32,40);(9,12)(10,16)(11,20)(13,22)(14,23)(15,18)(17,24)(19,21)(25,27)(26,30)(28,37)(29,38)(31,35)(32,36)(33,34)(39,40);(9,11)(10,17)(12,19)(13,18)(14,21)(15,24)(16,22)(20,23)(25,30)(26,35)(27,36)(28,34)(29,33)(31,32)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,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, 8004292);