#469691

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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)(5,6,11,16)(7,17,18,30)(8,21,22,29)(9,15,10,14)(12,25,26,20)(13,23,28,24)(19,27,32,35)(31,34,36,33);(7,18)(9,10)(12,26)(14,15)(19,32)(21,29)(23,24)(27,35);(1,2,3,4)(5,7,9,20,11,18,10,25)(6,12,14,17,16,26,15,30)(8,19,23,33,22,32,24,34)(13,27,29,36,28,35,21,31);(5,8)(6,13)(7,19,18,32)(9,24)(10,23)(11,22)(12,27,26,35)(14,21)(15,29)(16,28)(17,31,30,36)(20,34,25,33)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 376887);