#7720340

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	#37418	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4436 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)(9,10,14,16)(11,18,12,15)(13,21,22,17)(19,23,20,24)(26,31)(29,33)(32,38)(35,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,16,18)(13,19,22,20)(17,23,21,24)(25,26)(27,32)(28,29)(30,31)(33,37)(34,35)(36,38)(39,40);(10,16)(11,12)(17,21)(19,20)(26,31)(28,37)(32,38)(34,40);(1,3,4,7)(2,5,6,8)(9,12,14,11)(10,15,16,18)(13,20,22,19)(17,23,21,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,33,39);(9,10)(11,18)(12,15)(13,17)(14,16)(19,24)(20,23)(21,22)(25,28)(26,33)(27,34)(29,31)(30,37)(32,39)(35,38)(36,40);(9,13,14,22)(10,17,16,21)(11,19,12,20)(15,23,18,24)(25,29,30,33)(26,28,31,37)(27,35,36,39)(32,34,38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627536);