#7973021

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7880217);