#7932037

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^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, 7839233);