#206859

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,d^c=di

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, 114055);