#151876

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₃₂ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

266 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234280, #72234282
maximal normal subgroup of	#72234280, #72234282
maximal quotient of	#72234277, #72234287, #72234363

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59072);