#7712333

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	#38922	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=h,e^2=g,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, 7619529);