#7977460

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	#40525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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)(3,5,7,8)(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26,31,30)(27,29,36,32)(28,37,39,33)(34,38,35,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27,31,36)(26,32,30,29)(28,35,39,34)(33,40,37,38);(1,3,4,7)(2,5,6,8)(10,15)(12,17)(16,22)(20,23)(25,28,31,39)(26,33,30,37)(27,34,36,35)(29,40,32,38);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,40)(34,36);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29)(26,27)(28,38)(30,36)(31,32)(33,35)(34,37)(39,40);(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,30,31,26)(27,32,36,29)(28,37,39,33)(34,38,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,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, 7884656);