#464071

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 371267);