#492817

order	512
nilpotency class	4
isoclinism	isoclinic class of #492512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

473 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226856, #72226906, #72226947
maximal normal subgroup of	#72226856, #72226906, #72226947
maximal quotient of	#72226848

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400013);