#7917487

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=hi,e^2=i,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, 7824683);