#7710991

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

4302 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=i,f^2=g,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, 7618187);