#501152

order	512
nilpotency class	4
isoclinism	isoclinic class of #501150
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24404
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1109 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1109 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408348);