#7712777

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×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619973);