#148789

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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)(7,16)(8,21)(9,26)(10,15)(11,19)(12,24)(13,31)(14,23)(17,18)(20,30)(22,29)(25,33)(27,32)(28,35)(34,36);(5,6)(7,17,18,16)(8,21)(9,26)(10,15)(11,19,28,35)(12,24)(13,31)(14,23)(20,32,27,30)(22,34,36,29)(25,33);(5,7,9,27,10,18,23,20)(6,11,14,34,15,28,26,29)(8,22,24,35,25,36,31,19)(12,30,21,17,13,32,33,16)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240831, #72249975, #72250010, …
maximal normal subgroup of	#72240831, #72249975, #72250010, …
maximal quotient of	#72240836

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55985);