#148367

order	512
nilpotency class	4
isoclinism	isoclinic class of #13400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 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)(6,12,14,24)(8,20)(9,23)(10,22)(13,29,31,27)(15,34,33,28)(16,35,30,32)(17,21)(18,26)(25,36);(5,6,8,12,10,15,20,28,11,14,9,24,22,33,23,34)(7,16,17,35,18,29,21,13,19,30,36,32,26,27,25,31);(2,4)(5,7)(6,13)(8,21)(9,25)(10,26)(11,19)(12,27)(14,31)(15,35)(16,34)(17,20)(18,22)(23,36)(24,29)(28,30)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224865, #72224936, #72224942, …
maximal normal subgroup of	#72224865, #72224936, #72224942, …
maximal quotient of	#72224859, #72237230, #72280223

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55563);