#147289

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1919
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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:

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

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,g^a=gi,h^a=hi,d^b=dh,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	#72314250, #72314278
maximal normal subgroup of	#72314250, #72314278

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54485);