#7718429

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	#44880	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4489 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=i,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, 7625625);