#454973

order	512
nilpotency class	3
isoclinism	isoclinic class of #454970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,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, 362169);