#486706

order	512
nilpotency class	4
isoclinism	isoclinic class of #486291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

802 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=ghi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265442, #72265553
maximal normal subgroup of	#72265442, #72265553

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393902);