#395623

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2107 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=fi,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, 302819);