#7940404

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=h,f^2=i,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, 7847600);