#449205

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^c=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, 356401);