#7656527

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=gh,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563723);