#7719081

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	#45159	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=h,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, 7626277);