#212023

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

682 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)(3,6)(4,5)(7,8)(9,10,12,15,13,11,14,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,31)(28,30)(29,32);(2,5)(6,8)(17,19,22,29)(18,23,26,32)(20,27,21,28)(24,30,25,31);(1,3,4,7)(2,6,5,8)(18,24)(20,21)(23,30)(25,26)(27,28)(31,32);(9,11,12,16,13,10,14,15)(17,20,22,21)(18,25,26,24)(19,27,29,28)(23,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119219);