#469771

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 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:

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

Polycyclic presentation:

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

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, 376967);