#7972900

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4268 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)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26)(27,36)(28,34)(29,39)(30,31)(32,35)(33,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27)(26,33)(28,30)(29,37)(31,39)(32,38)(34,40)(35,36);(10,12)(16,17)(18,20)(23,24)(25,28,32,29)(26,34,35,39)(27,37,38,30)(31,36,40,33);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,29,32,28)(26,34,35,39)(27,37,38,30)(31,33,40,36);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,30,32,37)(26,31,35,40)(27,28,38,29)(33,39,36,34);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,31)(26,30)(27,34)(28,36)(29,33)(32,40)(35,37)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7880096);