#8093132

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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)(9,10,15,18)(11,16,13,12)(14,17,22,24)(19,23,21,20)(25,26)(27,34)(28,37)(29,31)(30,40)(32,36)(33,39)(35,38);(2,5)(6,8)(9,11,15,13)(10,12,18,16)(14,21,22,19)(17,23,24,20)(25,27,31,36)(26,32,29,34)(28,38,39,30)(33,35,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21)(25,28)(26,33)(27,30)(29,37)(31,39)(32,40)(34,35)(36,38);(9,13,15,11)(10,16,18,12)(14,21,22,19)(17,23,24,20)(25,29,31,26)(27,32,36,34)(28,37,39,33)(30,40,38,35);(1,2)(3,6)(4,5)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,27)(26,34)(28,30)(29,32)(31,36)(33,35)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,30)(26,35)(27,28)(29,40)(31,38)(32,37)(33,34)(36,39)

Polycyclic presentation:

2^9|a^2=gh,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000328);