#148855

order	512
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2088
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 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)(5,6,11,16,28,14,10,15,12,17,29,33,9,8,25,27)(7,18,23,31,22,19,21,34,24,26,36,35,20,13,30,32);(5,7,9,20,12,24,28,22)(6,13,14,18,17,19,8,26)(10,21,11,23,25,30,29,36)(15,31,33,34,27,35,16,32);(2,4)(5,8,9,6,12,14,28,17)(7,19,20,26,24,13,22,18)(10,27,11,16,25,15,29,33)(21,34,23,35,30,32,36,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250772, #72250780, #72250785, …
maximal normal subgroup of	#72250772, #72250780, #72250785, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56051);