#126653

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
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	#106
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#106
prime radical	C₂⁴	#106
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

490 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)(5,6,7,9,8,10,11,12)(13,14)(15,26,29,39)(16,22)(17,34)(18,35)(19,36)(20,25)(21,28,38,30)(23,33)(24,32)(27,40,31,44)(37,42,41,43);(2,4)(13,15,18,30,19,31,32,42,20,29,17,28,16,27,33,43)(14,21,24,39,22,37,35,44,25,38,23,26,36,41,34,40);(13,16,20,19)(14,22,25,36)(15,27,29,31)(17,32,18,33)(21,37,38,41)(23,34,24,35)(26,40,39,44)(28,42,30,43)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321367, #72321373, #72321374, …
maximal normal subgroup of	#72321367, #72321373, #72321374, …
maximal quotient of	#72322549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33849);