#371305

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 278501);