#147305

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250430, #72250433, #72250445, …
maximal normal subgroup of	#72250430, #72250433, #72250445, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54501);