#7933388

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	#40318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 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)(3,5,7,8)(9,10,15,17)(11,19,20,16)(12,22,23,14)(13,24,21,18)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,16,17,19)(12,21,23,13)(14,18,22,24);(10,17)(11,20)(12,23)(18,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(9,12)(10,14)(11,21)(13,20)(15,23)(16,18)(17,22)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3,4,7)(2,5,6,8)(9,13,15,21)(10,18,17,24)(11,12,20,23)(14,16,22,19)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,14)(10,12)(11,18)(13,19)(15,22)(16,21)(17,23)(20,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7840584);