#7895668

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=gi,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, 7802864);