#461543

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16395
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368739);