#218221

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125417);