#397249

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^d=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, 304445);