#416286

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=h,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,f^b=fi,e^c=ei,f^c=fi,g^c=gi

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, 323482);