#7779398

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	#34741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,24);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,17,13)(14,19,22,20)(18,23,24,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,3,4,7)(2,5,6,8)(10,17)(11,12)(14,22)(21,23)(26,31)(29,39)(32,37)(33,40);(9,12,15,11)(10,16,17,13)(14,20,22,19)(18,23,24,21)(25,27)(26,32)(28,34)(29,33)(30,36)(31,37)(35,38)(39,40);(9,13,15,16)(10,12,17,11)(14,21,22,23)(18,20,24,19)(25,28)(26,33)(27,35)(29,37)(30,38)(31,40)(32,39)(34,36);(9,14,15,22)(10,18,17,24)(11,19,12,20)(13,21,16,23)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,37,38,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7686594);