#217430

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

543 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)(4,6)(5,8)(10,15)(11,18)(12,14)(16,23)(19,24)(20,21)(25,26,29,33,30,34,28,32)(27,35,36,39,38,40,37,31);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,19,21,16)(12,22,14,17)(18,23,20,24);(3,8)(5,7)(9,11,14,18,13,21,12,20)(10,16,17,24,15,19,22,23)(25,27,28,37,30,38,29,36)(26,31,33,35,34,39,32,40);(2,6)(5,7)(9,12,13,14)(10,17,15,22)(11,20,21,18)(16,24,19,23)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,35,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,d^c=di

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, 124626);