#402445

order	512
nilpotency class	3
isoclinism	isoclinic class of #402444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14842
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

974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339024, #72339032, #72339037
maximal normal subgroup of	#72339024, #72339032, #72339037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309641);