#449383

order	512
nilpotency class	3
isoclinism	isoclinic class of #449383
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15941
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

845 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)(4,17)(5,20)(6,12)(7,11)(8,16)(9,23)(10,14)(15,26)(18,25)(19,31)(21,24)(22,27)(28,29)(30,32);(1,3,14,18,7,17,6,16)(2,8,12,4,11,25,10,13)(5,21,22,31,23,30,29,15)(9,26,27,32,20,19,28,24);(1,4,6,8,7,13,14,25)(2,3,12,18,11,17,10,16)(5,19,22,24,23,26,29,32)(9,15,28,30,20,31,27,21);(1,5,7,23)(2,9)(3,15)(4,19,13,26)(6,22,14,29)(8,24,25,32)(10,27)(11,20)(12,28)(16,21)(17,31)(18,30)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338910, #72338917, #72338919
maximal normal subgroup of	#72338910, #72338917, #72338919

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356579);