#8093696

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

4654 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000892);