#497976

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405172);