#497698

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
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×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404894);