#8095965

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

4627 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=gh,f^2=g,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, 8003161);