#7799522

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

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,e^b=ei,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, 7706718);