#495044

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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,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	#72239492, #72250530, #72250536, …
maximal normal subgroup of	#72239492, #72250530, #72250536, …
maximal quotient of	#72319243, #72319878

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402240);