#7654529

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4349 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=gi,f^2=gi,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, 7561725);