#446377

order	512
nilpotency class	3
isoclinism	isoclinic class of #423438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15841
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338910, #72338916, #72338917
maximal normal subgroup of	#72338910, #72338916, #72338917

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353573);