#7744644

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	#37276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7651840);