#8109901

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

4406 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)(3,7)(13,21)(17,22)(18,23)(20,24)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,2,4,5)(3,6,7,8)(10,15)(12,19)(17,22)(20,24)(28,36)(32,38)(35,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,28,31,36)(26,32,30,38)(27,35,33,39)(29,37,34,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8017097);