#7646303

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4468 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)(3,7,8,5)(9,10,15,17)(11,18,13,16)(12,14,22,24)(19,21,20,23)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(1,3,4,8)(2,5,6,7)(9,11,15,13)(10,16,17,18)(12,20,22,19)(14,21,24,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(9,12)(10,14)(11,19)(13,20)(15,22)(16,23)(17,24)(18,21)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(9,12)(10,14)(11,20)(13,19)(15,22)(16,21)(17,24)(18,23)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(9,13,15,11)(10,16,17,18)(12,20,22,19)(14,23,24,21)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,14)(10,12)(11,21)(13,23)(15,24)(16,20)(17,22)(18,19)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)

Polycyclic presentation:

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

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, 7553499);