#7865069

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

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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=g,f^2=h,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, 7772265);