#7898497

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	#35571	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4516 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)(9,10,15,17)(11,16,20,23)(12,22,18,13)(14,19,21,24)(25,26)(27,30)(28,34)(29,37)(31,33)(32,36)(35,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,18)(15,20)(17,23)(22,24)(25,27,32,31)(26,33,36,30)(28,37,39,35)(29,34,38,40);(10,17)(11,20)(12,18)(19,24)(25,28,32,39)(26,34,36,40)(27,35,31,37)(29,30,38,33);(1,4,5,2)(3,7,8,6)(9,12,15,18)(10,13,17,22)(11,21,20,14)(16,19,23,24)(25,29,32,38)(26,35,36,37)(27,34,31,40)(28,30,39,33);(9,13)(10,18)(11,19)(12,17)(14,16)(15,22)(20,24)(21,23)(25,30)(26,27)(28,38)(29,39)(31,36)(32,33)(34,35)(37,40);(9,14,15,21)(10,19,17,24)(11,18,20,12)(13,23,22,16)(25,31,32,27)(26,33,36,30)(28,37,39,35)(29,40,38,34)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,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, 7805693);