#403905

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14884
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

863 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311101);