#362739

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^c=hi,h^d=hi

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, 269935);