#395946

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1415 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:

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

Polycyclic presentation:

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

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, 303142);