#126785

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,d^b=di,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321374, #72321389, #72321392, …
maximal normal subgroup of	#72321374, #72321389, #72321392, …
maximal quotient of	#72322578

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33981);