#397945

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1832 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)(6,12)(7,17)(8,22)(9,10)(13,26)(14,29)(15,16)(19,33)(20,21)(23,24)(27,36)(32,35);(1,2,3,4)(5,6,9,12,11,16,10,15)(7,18,20,13,21,28,17,26)(8,14,22,25,24,29,23,30)(19,32,34,27,33,36,31,35);(5,7,9,17,11,21,10,20)(6,13,12,18,16,26,15,28)(8,19,22,31,24,33,23,34)(14,27,25,32,29,35,30,36);(5,8)(6,14)(7,19)(9,22)(10,23)(11,24)(12,25)(13,27)(15,30)(16,29)(17,31)(18,32)(20,34)(21,33)(26,35)(28,36)

Polycyclic presentation:

2^9|b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,g^c=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, 305141);