#7659032

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	#37468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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)(11,18,13,16)(12,17,22,21)(19,24,20,23)(26,31)(27,35)(33,34)(36,37);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,14,18)(12,20,22,19)(17,24,21,23)(25,26,30,31)(27,32,35,38)(28,33,29,34)(36,39,37,40);(9,11,15,13)(10,16,14,18)(12,19,22,20)(17,23,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,27,30,35)(26,32,31,38)(28,36,29,37)(33,39,34,40);(9,13,15,11)(10,16,14,18)(12,20,22,19)(17,23,21,24)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,18,13,16)(12,21,22,17)(19,24,20,23)(25,29,30,28)(26,34,31,33)(27,37,35,36)(32,40,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566228);