#7712262

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	#37379	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4407 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,e^2=h,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, 7619458);