#7734360

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	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7641556);