#135802

order	512
nilpotency class	4
isoclinism	isoclinic class of #135783
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1844
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#498
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=e,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cgi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287034, #72287170, #72287173, …
maximal normal subgroup of	#72287034, #72287170, #72287173, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42998);