#7972319

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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)(9,10,15,17)(11,21,22,16)(12,14,20,18)(13,23,24,19)(26,31)(27,35)(33,40)(34,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,19)(13,18)(14,24)(15,22)(17,21)(20,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,39)(35,38)(36,40);(1,3,5,4)(2,7,6,8)(10,17)(11,22)(12,20)(13,24)(26,31)(28,36)(32,38)(34,37);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,19,22,23)(13,21,24,16)(25,27,30,35)(26,32,31,38)(28,34,36,37)(29,33,39,40);(9,13,15,24)(10,19,17,23)(11,18,22,14)(12,21,20,16)(25,28)(26,33)(27,34)(29,38)(30,36)(31,40)(32,39)(35,37);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,13,22,24)(16,19,21,23)(25,29,30,39)(26,34,31,37)(27,33,35,40)(28,32,36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7879515);