#148365

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dh,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237224, #72237285, #72238781, …
maximal normal subgroup of	#72237224, #72237285, #72238781, …
maximal quotient of	#72237234

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55561);