#7736383

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	#51424	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=ghi,e^2=h,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, 7643579);