#469770

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1028 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)(5,6,10,14)(7,11,17,27)(8,18,20,12)(9,22,24,13)(15,30,23,25)(16,26,19,28)(21,29,34,32)(31,36,33,35);(7,15,17,23)(11,25,27,30)(16,31,19,33)(26,35,28,36);(1,2,3,4)(5,7)(6,11)(8,19,20,16)(9,23)(10,17)(12,28,18,26)(13,30)(14,27)(15,24)(21,31,34,33)(22,25)(29,35,32,36);(5,8,9,21,10,20,24,34)(6,12,13,29,14,18,22,32)(7,16,23,33,17,19,15,31)(11,26,30,36,27,28,25,35)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 376966);