#151879

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 236 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210680, #72210682
maximal normal subgroup of	#72210680, #72210682
maximal quotient of	#72234237, #72234277

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59075);