#492754

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

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

normal subgroups only

This group has 445 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=gi,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	#72226906, #72226945, #72226954
maximal normal subgroup of	#72226906, #72226945, #72226954
maximal quotient of	#72226877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399950);