#151041

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₁₆	D₄
commutator	C₄	C₂²×C₃₂
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×C₄⋊C₁₆
prime radical	C₂³	C₄⋊C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

172 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58237);