#467076

order	512
nilpotency class	3
isoclinism	isoclinic class of #464719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319592, #72319923, #72320049
maximal normal subgroup of	#72319592, #72319923, #72320049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374272);