#7712802

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	#37455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4596 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7619998);