#7919982

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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)(9,10,15,17)(11,20,23,16)(12,18,24,14)(13,21,22,19)(25,26)(27,38)(28,35)(29,40)(30,33)(31,36)(32,37)(34,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,21)(13,18)(14,22)(15,23)(17,20)(19,24)(25,27)(26,31)(28,29)(30,39)(32,36)(33,40)(34,35)(37,38);(1,3,5,4)(2,7,6,8)(10,17)(12,24)(16,20)(19,21)(25,28,32,30)(26,33,37,35)(27,39,36,29)(31,34,38,40);(9,12)(10,18)(11,21)(13,16)(14,17)(15,24)(19,23)(20,22)(25,29,32,39)(26,34,37,40)(27,28,36,30)(31,35,38,33);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,19,17,21)(11,18,23,14)(12,16,24,20)(25,30,32,28)(26,35,37,33)(27,39,36,29)(31,34,38,40);(9,14,15,18)(10,12,17,24)(11,22,23,13)(16,21,20,19)(25,31,32,38)(26,36,37,27)(28,34,30,40)(29,35,39,33)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=h,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, 7827178);