#7919932

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 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)(9,10,15,12)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,26,31,34)(27,35,38,32)(28,29,39,40)(30,33,36,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23)(25,27)(26,32)(28,30)(29,37)(31,38)(33,40)(34,35)(36,39);(1,3,5,4)(2,7,6,8)(10,12)(16,19)(17,23)(18,24)(25,28)(26,29)(27,36)(30,38)(31,39)(32,33)(34,40)(35,37);(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,24,20,18)(25,29)(26,28)(27,37)(30,32)(31,40)(33,38)(34,39)(35,36);(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,24,19,18)(25,30,31,36)(26,33,34,37)(27,28,38,39)(29,32,40,35);(1,4,5,3)(2,8,6,7)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,23,19,17)(25,28)(26,29)(27,30)(31,39)(32,37)(33,35)(34,40)(36,38)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=h,e^2=hi,f^2=gi,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, 7827128);