#7732320

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	#37277	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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)(9,10,16,15)(11,21,23,17)(12,18,13,19)(14,20,22,24)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,17)(12,14)(13,22)(15,21)(16,23)(18,24)(19,20)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,3,5,4)(2,7,6,8)(9,12,16,13)(10,18,15,19)(11,22,23,14)(17,20,21,24)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39);(1,4,5,3)(2,8,6,7)(9,13,16,12)(10,19,15,18)(11,22,23,14)(17,20,21,24)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39);(9,14,16,22)(10,20,15,24)(11,12,23,13)(17,19,21,18)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,15,16,10)(11,17,23,21)(12,19,13,18)(14,24,22,20)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

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, 7639516);