#219020

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	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 126216);