#512043

order	512
nilpotency class	4
isoclinism	isoclinic class of #511839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#32976
prime radical	#3483	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

775 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,d^c=di,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321074, #72342994, #72343000
maximal normal subgroup of	#72321074, #72342994, #72343000

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419239);