#148131

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
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

698 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)(5,6,9,18,12,19,29,15)(7,20,21,13,24,8,23,27)(10,30,11,16,28,17,34,36)(14,32,22,25,35,26,31,33);(5,7,9,21,12,24,29,23)(6,13,15,20,19,27,18,8)(10,31,11,14,28,22,34,35)(16,25,30,32,36,33,17,26);(2,4)(5,8)(6,14)(7,16)(9,13)(10,32)(11,33)(12,20)(15,22)(17,23)(18,31)(19,35)(21,30)(24,36)(25,34)(26,28)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249886, #72249891, #72249944, …
maximal normal subgroup of	#72249886, #72249891, #72249944, …
maximal quotient of	#72264842

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55327);