#7711485

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

4298 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)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(27,30)(29,37)(32,34)(33,36);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,21,24,23)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(10,17)(12,18)(16,19)(20,22)(25,27,31,30)(26,32,28,34)(29,38,37,35)(33,40,36,39);(9,12,15,18)(10,13,17,24)(11,20,14,22)(16,21,19,23)(25,28,31,26)(27,32,30,34)(29,36,37,33)(35,39,38,40);(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,22)(19,20)(25,29,31,37)(26,33,28,36)(27,35,30,38)(32,39,34,40);(1,3,4,7)(2,6,5,8)(9,14,15,11)(10,19,17,16)(12,22,18,20)(13,23,24,21)(25,30,31,27)(26,32,28,34)(29,38,37,35)(33,39,36,40)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=i,f^2=gh,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, 7618681);