#148134

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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

702 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)(5,6,12,18)(7,8,23,27)(9,19,13,16)(10,30,11,20)(14,34,24,21)(15,26,32,25)(17,28,35,29)(22,31,36,33);(5,7,9,21,12,23,13,34)(6,14,16,8,18,24,19,27)(10,15,28,22,11,32,29,36)(17,26,20,33,35,25,30,31);(5,8)(6,15)(7,20)(9,24)(10,31)(11,33)(12,27)(13,14)(16,22)(17,34)(18,32)(19,36)(21,35)(23,30)(25,29)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250359, #72250362, #72250376, …
maximal normal subgroup of	#72250359, #72250362, #72250376, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55330);