#495045

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319775, #72319789, #72319844, …
maximal normal subgroup of	#72319775, #72319789, #72319844, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402241);