#7917805

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	#43184	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 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)(12,16)(13,15)(14,19)(18,23)(20,22)(21,24)(25,26,32,38)(27,31,37,33)(28,36,30,34)(29,40,39,35);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,33)(28,29)(30,39)(31,38)(32,37)(34,40)(35,36);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,22)(21,23)(25,28,32,30)(26,34,38,36)(27,39,37,29)(31,40,33,35);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,22)(23,24)(25,29,32,39)(26,35,38,40)(27,28,37,30)(31,34,33,36);(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,30,32,28)(26,36,38,34)(27,39,37,29)(31,40,33,35);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)(25,31)(26,37)(27,38)(28,40)(29,34)(30,35)(32,33)(36,39)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=i,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, 7825001);