#416287

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=hi,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, 323483);