#149091

order	512
nilpotency class	4
isoclinism	isoclinic class of #135930
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	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

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

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, 56287);