#7712219

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4359 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=gh,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, 7619415);