#149510

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₃₂ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

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

200 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 158 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:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(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);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234277, #72234278, #72234279
maximal normal subgroup of	#72234277, #72234278, #72234279
maximal quotient of	#72234280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56706);