#7798054

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	#50677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 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)(9,10,15,12)(11,16,18,14)(13,17,23,20)(19,24,21,22)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,22,20,24)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(1,3,5,4)(2,6,8,7)(11,18)(14,16)(19,21)(22,24)(26,33)(28,34)(32,36)(35,40);(1,4,5,3)(2,7,8,6)(9,12,15,10)(11,14,18,16)(13,17,23,20)(19,24,21,22)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(9,13,15,23)(10,17,12,20)(11,19,18,21)(14,22,16,24)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,21)(23,24)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7705250);