#7654211

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4405 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^2=g,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, 7561407);