#7654985

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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)(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27,31,39)(26,32,36,37)(28,35,34,38)(29,30,33,40);(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,28,31,34)(26,33,36,29)(27,38,39,35)(30,37,40,32);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,29,31,33)(26,34,36,28)(27,30,39,40)(32,38,37,35);(9,13,15,10)(11,17,12,16)(14,18,22,21)(19,23,20,24)(25,26)(27,32)(28,33)(29,34)(30,38)(31,36)(35,40)(37,39);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30)(26,35)(27,33)(28,37)(29,39)(31,40)(32,34)(36,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562181);