#8093512

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

4458 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)(9,10)(11,16)(12,17)(13,20)(14,15)(18,23)(19,22)(21,24)(25,26,31,36)(27,34,29,32)(28,33,30,35)(37,40,38,39);(2,5)(3,7)(11,19)(12,21)(16,22)(17,24)(25,27,31,29)(26,32,36,34)(28,37,30,38)(33,39,35,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,19)(17,24)(18,20)(25,28,31,30)(26,33,36,35)(27,37,29,38)(32,39,34,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,17)(14,19)(15,22)(18,21)(23,24)(25,29,31,27)(26,34,36,32)(28,37,30,38)(33,39,35,40);(9,12,14,21)(10,17,15,24)(11,20,19,18)(13,22,23,16)(25,30,31,28)(26,35,36,33)(27,37,29,38)(32,39,34,40);(9,13)(10,18)(11,17)(12,22)(14,23)(15,20)(16,21)(19,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=ghi,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, 8000708);