#8075318

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

4628 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)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(1,2)(3,6)(4,7)(5,8)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(10,15)(13,22)(16,23)(20,24)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(1,3,5,4)(2,6,8,7)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7982514);