#149079

order	512
nilpotency class	4
isoclinism	isoclinic class of #135928
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 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:

(5,6,9,15,12,16,29,24,10,17,11,30,19,35,31,36)(7,18,21,34,22,8,14,13,23,27,20,33,32,26,28,25);(1,2)(3,4)(5,7,19,22,10,23,12,32)(6,13,16,34,17,25,35,33)(8,24,18,30,26,36,27,15)(9,28,31,21,11,14,29,20);(2,4)(5,8,9,25,12,27,29,34,10,26,11,13,19,18,31,33)(6,14,15,7,16,20,24,22,17,28,30,23,35,21,36,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56275);