#7917730

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

4204 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2)(3,7)(4,8)(5,6)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gi,e^2=g,f^2=ghi,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, 7824926);