#512040

order	512
nilpotency class	4
isoclinism	isoclinic class of #511842
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	#3481	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 755 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419236);