#8093406

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
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4572 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000602);