#7736392

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	#44270	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4449 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:

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

Polycyclic presentation:

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

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, 7643588);