#7799520

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

4262 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)(3,7,8,5)(9,10,15,18)(11,19,14,16)(12,13,17,24)(20,21,22,23)(25,26)(27,31)(28,30)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3,4,8)(2,5,6,7)(9,11,15,14)(10,16,18,19)(12,20,17,22)(13,23,24,21)(26,28)(31,33)(32,36)(38,39);(11,14)(16,19)(20,22)(21,23)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(9,12)(10,13)(11,20)(14,22)(15,17)(16,23)(18,24)(19,21)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,13)(10,17)(11,21)(12,18)(14,23)(15,24)(16,20)(19,22)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,14,15,11)(10,16,18,19)(12,22,17,20)(13,23,24,21)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7706716);