#501145

order	512
nilpotency class	4
isoclinism	isoclinic class of #501144
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

1078 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1078 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408341);