#148509

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

680 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)(5,6,10,16,12,17,11,32)(7,18,21,23,22,34,33,24)(8,20,25,36,26,19,13,28)(9,27,29,15,31,35,30,14);(5,7)(6,13)(8,16)(9,28)(10,21)(11,33)(12,22)(14,24)(15,23)(17,25)(18,27)(19,30)(20,29)(26,32)(31,36)(34,35);(2,4)(5,8,9,23,10,25,29,34,12,26,31,24,11,13,30,18)(6,7,14,19,32,33,35,36,17,22,15,20,16,21,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249994, #72250004, #72250010, …
maximal normal subgroup of	#72249994, #72250004, #72250010, …
maximal quotient of	#72308751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55705);