#206863

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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

860 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)(12,16)(13,23)(14,17)(15,21)(19,20)(22,24)(25,26,29,33,30,34,28,32)(27,35,38,39,36,40,37,31);(2,5)(6,7)(9,11,13,21,14,22,12,20)(10,15,16,18,17,19,23,24)(25,27)(26,31,34,39)(28,37)(29,38)(30,36)(32,40,33,35);(1,3,4,8)(2,6,5,7)(9,10)(11,19)(12,23)(13,16)(14,17)(15,22)(18,20)(21,24)(27,36)(31,39)(35,40)(37,38);(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,18,19,24)(25,28,30,29)(26,32,34,33)(27,37,36,38)(31,40,39,35)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^c=dhi

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, 114059);