#451252

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15991
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

740 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358448);