#7712414

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

4356 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)(9,10,15,12)(11,19,22,16)(13,17,21,23)(14,24,20,18)(25,26)(27,35)(28,32)(29,38)(30,36)(31,34)(33,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23)(25,26,31,34)(27,35,38,29)(28,32,40,37)(30,33,39,36);(1,2,5,4)(3,6,8,7)(9,10,15,12)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,27,31,38)(26,29,34,35)(28,33,40,36)(30,37,39,32);(1,2,5,4)(3,6,8,7)(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,24,20,18)(25,28,31,40)(26,32,34,37)(27,36,38,33)(29,39,35,30);(1,4,5,2)(3,7,8,6)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,24,19,18)(25,29)(26,27)(28,39)(30,40)(31,35)(32,36)(33,37)(34,38);(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,23,19,17)(25,30,31,39)(26,33,34,36)(27,37,38,32)(28,29,40,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=g,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, 7619610);