#7883079

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4511 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7790275);