#7653087

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gi,f^2=i,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, 7560283);