#462156

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369352);