#460864

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318947, #72318981, #72319003
maximal normal subgroup of	#72318947, #72318981, #72319003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368060);