#8095810

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

4575 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)(3,4)(5,7)(6,8)(9,10)(11,13)(12,17)(14,15)(16,18)(19,22)(20,24)(21,23)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(10,15)(11,18)(17,23)(19,24)(28,36)(31,38)(34,40)(37,39);(9,11)(10,16)(12,19)(13,15)(14,18)(17,20)(21,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,20)(21,24)(22,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,20)(18,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,3)(2,4)(5,8)(6,7)(9,13,14,16)(10,11,15,18)(12,20,21,22)(17,24,23,19)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

2^9|e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003006);