#499361

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#481
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=i,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi

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, 406557);