#7642879

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	#36818	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550075);