#490505

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

553 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 553 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225448, #72225507, #72225511
maximal normal subgroup of	#72225448, #72225507, #72225511
maximal quotient of	#72225429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397701);