#492451

order	512
nilpotency class	4
isoclinism	isoclinic class of #492252
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
prime radical	#2246	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(4,12)(7,13)(8,15)(9,16)(10,14)(17,18)(19,26)(20,31)(21,27)(22,36)(23,24)(25,34)(28,38)(29,45)(30,43)(32,44)(33,47)(35,40)(37,42)(39,48)(41,46);(1,2,5,7)(3,9,11,15)(4,13,12,6)(8,14,16,10)(17,19,21,23)(18,24,27,26)(20,29,33,32)(22,30,35,38)(25,39,41,42)(28,40,43,36)(31,44,47,45)(34,37,46,48);(2,7)(6,13)(8,16)(9,15)(17,20,21,33)(18,25,27,41)(19,29,23,32)(22,37,35,48)(24,39,26,42)(28,44,43,45)(30,46,38,34)(31,40,47,36);(1,3,5,11)(2,8,7,16)(4,14,12,10)(6,15,13,9)(17,19,21,23)(18,26,27,24)(20,32,33,29)(22,30,35,38)(25,39,41,42)(28,36,43,40)(31,44,47,45)(34,48,46,37)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399647);