#7719060

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	#50865	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626256);